The Vijnana Parishad .of India

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:\ ... - ·) .

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Published by :

The Vijnana Parishad .of India DAY ANAND VEDIC POSTGRADUATE COLLEGE

Bundelkhand University .

ORAi, U. P., INDIA

J:RANABHA

H. M. Srivastava r.Jniversity o.f Victoria Victoria,. B. · C., Canada

'

EDITORS

AND

··" ...

R. C. Singh Chandel D. V. Postgraduate College

Orai, U. P., India

EDITORIAL ADVISORY BOARO. Chief Advisor: J. N. Kapur (Kanpur)

R. G. Buschman (Laramie, WY) R. $. Mishra (Lucknow)

L. Carlitz (Durham, NC) L. S. Kothari (Delhi)

C; Prasad (Roorkee) B. E. Rhoades (Bloomington, IN)

H. K. Srivastava (Lucknow)

K. M. Saksena (Kanpur) S. P. Singh (St. John's) L. J. Slater (Cambridge, U.K.)

K. L. Chung (Stan.ford) R. C. Mehrotra (Jaipur) K. N. Sriv~stava (Bhopal) A. B. Tayler (Oxford, U.K.) V. K. Verma (Delhi)

Printed at: Navin Printing Press, Go pal Gatij, Orai-235001, U .P.,India

Jiianabha, Vol. 15, 1985

TIME DEPENDENT SLOW FLOW OF A VISCOUS

INCOMPRESSIBLE FLUID 8~TWEEN W AYY WALLS

By

D. M. Mehta and Y. N. Gaur

Department of Math~i:natics, M •. R. Engineering College,

Jaipur-'302017, Rajasthan, Indfa,

(R~ceived : October JO, 1982; Revised : May 9, 1984)

ABSTRACT

Sl~w- flow of a viscous incompressible fluid through rough parallel

plates, under the application of periodic pressure gradient, has been

considered. The expressions presented for the physical quantities

involved are true for all values of tJ1e. freql!enc;y (>[the pressure gradi

ent. The roughness is assumed to be small. The effect of the freque

ncy a;d iime on the v~locity . profiles is studied ;u~'~ri~.iny f~r the

particular- cas~s o'r si'nusoidal roughness when the phas~ differei1ce~are zero and n;.

l. IN'IRODUCTION

The exact solutions of Na vier-Stoke!>' equations for a viscous in

compressible fluid with axially parallel flow thr01.igh a tube of circular

cross-section, under the influence of periodic press~re gradie11t have

been studied by Sex! [ 8 ] and Uchida [ 9 ], while those of co-axial

circular cylinders by Verma [ 10 ]. Hepworth and Rice [ 4,5] have

also studied the flow between parallel plates and circular rectangula

tubes with arbitrary time varying pressure gradient. The slow viscous

flow between rotating concentric infinite cylinders with axial roughness

au . aw ·· p ·pa~ : .d. . tv U = -, rt= ----, X = x/a, Z = z/a, = ----an ·T=-, .

v v v2p a'.!.

The plates are symmetrically placed on the two sides of the x-axis,

with a distance 2a between them am:l• z is· measwred- at tigh.~, angles

to the plates. U, V,P and Tare non-dimensional longitudinal velocity,

transverse velocity, pressure and time respectively.

Under the assumption of slow motion,

(4) V2P=O.

Using (4) in (1), we have

(5) (--32 + (l2 )2 u = 1- ra2u + ·~~ )· ()X2 ()Z2 : oT \ cX2 cZ2 .

The boundary conditions are

(6) U = 0 = Wat Z = 1 -\- e1Y1 (X)

and Z = - I -t r:.N2 (X), T > O,X > 0,

where r:. < < is the roughness parameter and J'V1 (X) and Af2(X) are

arbitrary functions of X.

Let

(7) P(X, Z, T) = Po(X, T) -\- P1(X, Z, T),

WlX, Z, T) = W1(X, Z, T),

and

U(X, z. T) = Uo(Z; T; + U1(X, Z, T),

where Pi, U1 and W1 are the variations cause-d:hy th€ roughness and

4 )

Po a:nd Uo are the quantities for the case of smooth plates •.

Let ;·,.

(S) - fx0 = K cos nT = Re [KinT J,

and

(9) U0 = Re [ /(Z) /nT ],

where Re means 'the real part of' and K is. a constant.

The solution for Uo is given by

where m = if(in).

Form (!), (2). (3), (5) and (7), we have

fll) oUi = _ 0P1 + (a2u1 + 22u1_)

· aT @T · 2x2 azi /

{l3) 0U1 + 0W1 = O, oX cZ

and

. under- the boundary conditiops.

[ 5

(15) U1 = - Uo, W1 = 0 at Z = l+ e N1(X) and

Z = - 1 + e N2(X), T > 0, X > 0,

and

U1 = 0 at - l + e N2(X) < Z < 1 + e Ni(X), T > 0, X > 0.

3. METHOD OF SOLUTION

Following (9), we assume

T X inT (16) U1 (X, Z, ) =Re [U1 ( , Z) e ]

and therefore equation (14) reduces to

7 ( 32 32 )2 . (32U1 32r;1)

( l ) ax2 + 3z2 U 1 = m 2x2 + oz2 .

Let us suppose that

and

l JO°. ( 19) F('E,, Z) = (2/<t) 2 O f(X, Z) sin (~X) dX,

where F('E,, Z) is the Fourier sine transform of f(X, Z).

Using equation (18) in (17), taking the Fourier sine transform and

solving the resulting equation, \re get

(20) U('E,, Z) = A ('E,) e-bZ + B('E,) iZ + C('E,) e-~Z + D('E,) e'E,Z

where U('E, Z) is the Fourier sine transform of U1 (X, Z) and A(~),

6 ]

B(~), cm and D(~) are constants of integration and b = (~2 + in)1t2,

Taking inverse Fo!.l.rier sine transform of equation(2.0), we get

+ Dl~) e~Z J sin (~X) einT d~.

Using (21), in (13), we get

1 Joo -bZ (22) W1 (X,Z,T) =Re - (2/rt) 2 .

0 [- ~/b A(~) e

+ ~fb B(~) iZ - Cm e-~Z + D(~) e~z cos (~X)eint d~.

Using equation (21) and (22) in (11) and (12), we get

(23) Pi(X,Z,T) =Re (2rt)! in J~ !/~ {C(i;) e-~Z +nm e~z]cos(~X)

einT d~ + C,

where c is a constant or integration.

Let us assume

(24) A(~) = Ao(~) + e A1 (~) + ...

and similar expansions for B(~), cm and D(O.

Using the boundary conditions ( 15) in equations (21) and (22) and

equating the co-efficients of eO and e and inverting the resulting equa

tions by, Fourier sine and cosine integral theorems, we get

(25; A0 (X) = B0 (X) = Co(X) = Do(X) = 0,

Kb.:_ g(n) [ _l _ { N (X) - N (X) } x 4 G(X) .

1 2

I - -sinh X - H(X) {N1(X) + N2(X)} cosh X ],

B1(X) =Kb' g(n) [_I_ { N1(X) - N2(X)} X 4 G(X) .

sinh X + - 1-{ N1 (X) + N2 (X)} cosh X ], H(X) -

and

C1(X) = KXg(n) ( - _I_ { N1 (X) - N<>(X)} x 4 G(X) -

sinh b' + _I - { N1 (X) + N2(X)} cosh b' ], H(X)

- K Xg(n) [ --1- { N1 (X) - N2(X)} x 4 G(X) .

where G(X) = b' cosh b' sinh X - X cosh X sinli b',

H(X) b' sinh b' coshX - X cosh b' sinh X,

r 7

b 1 = (X2 + in)l/2 and g(n) is a known function of n, given by

g(n) = (I/vin). tanh (i1f2 nl12). N1 and N2 are Fourier sine transforms

of N1 and N2, respectively.

Thus the complete expressions for the velocities and pressure are

given by

(27) U(X, Z, T) =· U0 (Z, T) + U1(X, ~' T)

8 J

K co sh ( i112 n112 Z) inT = Re [ in ( 1 - cosh (il/2 nl/2) e )

-f- Re [ K/2 g(n) i: (2/Tt)l/2

J 00

[ _!!_ sin~ ; cosh bZ - ; sinh b_~sh ; ~ 0 b sinh ~ cosh b - ~ sinh b cosh ~

+ b cash_j_sinh bZ - ; cosh b sinh ; Z . {Ni (;) + 1'J'2 m } ]

b cosh ; sinh b - ; sinh b cosh 1;

sin (;X) /nT di; ],

(28) W (X, Z, T) = W1 (X, Z, T) = Re [ - K/2 g(n) i: (?./TC)l/2

Joo ~ [ si~h ~ sinh bZ - si~h b sinh ~ Z { Ni m + N

2 (6 }

0 b smh ~ cosh b - ~ smh b cosh ~

+ cosh ~ cosh bZ - cosh b cosh ~ Z r N (~) + :N2

(1;) } ] b cash ~ sinh b - ~ cash b sinh i; L

1

cosh (~X) einT di; ],

and

(29) P(X, Z, T) = Po (X, T) + P1 (X, Z, T)

= C - K X cos nT- Re [ K/2 g(n) e (2/1')1/2 in.

IO-:: [ sinh b cosh ~ .~. { Ni m _ :N2 (~) }

0 b sinh !; cash b -'- i; sinh b cash 1;. ·

+ cash b sinh ~Z _ _ b cash~ sinh b - ~cash b sinh ~ {N1 m + Nz m } ]

inT cash (~X) e d~ ],

where C is a constant of integratipn.

4. Particular Cases (Sinusoidal Jloughness)

Case (i) Let

(38) N1 (X) = sin Xjl = - N2 (X)

where 2nl is the wavelength of the roughness at the walls.

i. e. the phase difference in the roughness of the walls is n;.

We m,1y form· t1y write

(31) N1 m = - N 2 m = (rc/2)1/2. a (~ - If!),

[ 9

where N1 (~) and N2(~) are t!-ie Fourier sine transforms of N1(X) and

N2(X), respectively, and 1l is the Dirac delta function.

Substituting (31) in (27) and (28) and using the property of Dirac

delta functi<Jtl (Sneddon [7]), the longitudinal and the transverse velo

cities fur I = l and for different values of n are obtained as follows :

For n = I

(32) U(X, Z, T) [ U0 (Z, T),'=I - Re [Ke: [(I. 07626 + 2.46087 i)

cash {Z( 1 + i)l/2} - (0. 68334 + 2. @1039 i) cash Z] sin X. eiT ],

(33) W(X, Z, T) = Re [Ke: [ ( I. 07626 + 2. 46087 i)

sinh {Z(I +i)1/2} ---(I+ i)fi2-

IO j

- (0. 68334 + 2. 61039 i) sinh Z J cos X eiT].

For n ;= 2

(34) U(X, Z, T) = [ Uo(Z, T)]n_2 - Re [Ke (( 0. 76328 + 0. 84048 i)

cosh {Z (1+2 i)l/2} - (0. 47817 +I. 05276 i) cosh Z) sinX. e2iTJ,

(35) W(X, Z, T) =Re [Ke [ (0. 76328 + 0. 84048 i)

sinh {Z (I + 2i)li2} (l+2i)ll2--

- (0. 47817 + I. 05276 i) sinh Z] cos X. e2iT].

For n = 4

(36) U(X, Z, T) = [ Uo(Z, T) ],._4

- Re [KE [(O. 35991 + 0, 16441 i), cosh {Z (1+4i)l/2}

- (0. 21517 + 0. 36588 i) cosh Z) sin X e4iTJ,

(37) W(X, Z; T) = Re [KE [ (0. 35991 + O. 16141 i)

sinh {Z 0._=t-~i)l1 2 } -(0. 21517 + 0. 36588 i) sinhZ) coshX.e4"7]. -.-- (1 +4i)112

Case (ii)

Let

(38) N1(X) = N2(X) = sin (X/l),

i. e. the phase difference in the roughness of the two walls a zem.

The expression for U and W for l = I, now are

For n = 1,

[ 11

(39) U(X, Z, T) = [Uo(Z, T],.=1

- Re [K € [(O. lr265 + 1. 20l46 i). sinh {Z(l+i)l/2}

- (0. 11925 + 1. 05053 i) sinh ZJ sin X. eiTJ,

(40) W(X,Z, T) =Re [KE [(O. 11265 +I. 20! 46 i) cosh{Z_(l+i)ll2} (l+i)112

- (0. 11925 + I. 05053 i) cosh Z] cos X: eiT].

For n = 2,

{41) U(X, Z, T) = [Uo(Z, T) ]n-2

- Re [KE ((0 2045 l + 0. 59854 i) sinh {Z'.l + 2i)li2}

- (0. 08542 + 0. 48298 i) sink Z] sinX e2ffJ,

(42) W(X, Z, T) =Re [K € [(O. 20451 + o. 59851 i) cosh {Z(l+2i)112} (1 +2i)l/2

- (0. 08542 + 0. 48298 i) cosh ZJ cos X e2iT].

For n = 4

(43) U(X, Z, T} = [Uo(Z, TJn-4

- Re [f( E[(O. 19472 + 0. 20719 i). sinh {Z(1+4i)l/2}

-(0. 04171+0 21354 i)sinhZ] sinXe4iT). ·

(44) W(X, Z, T) = Re [ K E[(O. 19472 + 0. 20719 i)

co sh {Z (l + 4i)112} (1+4i)l/2 --

- (0. 04171 + 0.21354 i) cosh Z] cos X. e4•T].

12 ]

5. NUMERICAL DISCUSSIONS

The longitudinal velocity profiles for the wavy walls (phase

difference of the sinusoidal roughness of the walls being n) at different

cross-sections for o: = 0. 1, I = 1 and for different values of n and T

are shown by Figures I, 2 and 3. It is observed that the longitudinal

velocity decreases as the distance between the walls increases and vice

versa. For T = 0 and T = I, the increase in the frequency of the

applied pressure gradient decreases the magnitude of the longitudinal

velocity. For T = 2, the magnitude of the longitudinal velocity is

quite small for n = 1, which further decreases when n becomes 2, but

again increases as it becomes 4.

The transverse velocity profiles for the same values of paran~e

ters as above, at the plane X = 0 are depicted in Figures 4, 5 and 6.

For X ~ 1'/2 and 31'/2, the transverse velocity vanishes identically for

all values of Z and for X = 'TC, it is the image of the curves drawn for

X = 0. For T = 0 and T = 1, the increase in the frequency of the

applied pressurf' gradient decreases the transverse velocity. For the

forward pressure gradient, the resulting effect of the longitudinal and

the transverse velocities is that the direction of the flow is towards th~

boundary . when the width between the plates increases and vice

versa.

Figure 7 depicts the transverse velocity profiles for the same

values of e:, l, n and T, at the plane X = 0 when the phase difference

of the sinusoidal roughness of the walls i<i zero. It is noted that ~he

transverse velocity is optimum at the mid plane and for T = 0 and

T = 1, it decreases with the increase in the frequency ofappljed pres

sure gradient. The disturbances in the flow become more rapid with

the increase in the rqughness parameter e: .•

J

Fig. 1

[ 13

U/K \~ I'\ ~ 1\ O~\·\. 1\\ \\ \· \ ' I · I \

I \ I \ I \. i \ .\ I . I \ \ \ I '

I . t \ I . I i

o·s

I \ .. \ l " \ \ \ I ' \ . , ~ I I : \ I

I I ! I . i I X I _: ' . I I ·-'--L-. -,---i- ·-or1 r I lT/21 • I ,,. I . ) 1n1c J ! 2n I :

-o ·2 I I I . l . I I I . I i l I I ! f I I 11· II 1/ 1· I

·o·4 t I 1· 1 v· 1. J 1 / I I I . I I . I . I I I . -o ·6 I ; I I I / I 1· I I

I. I • I I .

• O·& t{I f / . ! /. !J/ I /

The longitudinal velocity profiles at different sections of a roughness wave taking l = 1, e; = 0. I_, and T = 0 for n = 1

{-), 2(-.-.-.-) and 4(- - - -).

, I I

... 1 11 I 1 I l I

~- L _j ~'- L i L ;__ I .L __, : l )(

Fig. 2

I IT/2. - --

l ! I I \ I \ I J

\1

The longitudinal velocity profiles at different sections of the

roughness wave tak,ing I= I, E = O. l, and T = 1 for n = l

(-), 2(-.-.-.-) and 4(- - - -).

MJ

U/K

_1·, ~/\ I I

i \ . I

I \. I I / I I \ I I I

I

I I \ i

Nl:'

I I I I I I i I

I I \ I

I I I I I I.

I ,

I I

J i I I

I I I I I

-' L-. x

Fig:3

I 0 I n12 I I

'II i \ I I

r )TI/ ,,. I

I I zn

I I I I I

f

I I I I I I

I I

\ ! I I I

I I I r \ I i i

I

' \ I

\ I \ I

\ I

I \ I \

\ I ·, . I ' I

. I I \ I

\

\ I

I

·.1 ,,, .,

The longitudinal velocity profiles at diffor,ent sections of a

roughness wave taking I= 1, .e = 0. I and T = 2 torn = 1 (.'..:..)~ 2(-:-. ..;.,'-)·and 4{·)_;.;,..·_). · · ·

z 1·0

.,I ' i ' ~ I . ' W/K •0·0~ -0·01 . 0 ~ O·Oi 0 02

-0·2J S·-..M' ·.· -0·411 '-;:\.· . \ -O·f; I I '

-oio·1~/ ,,..·/.' . /,,. ...,, +o ..

Fig ;i4.: : Tlie tMnswets-e vefod ty profiles · of a roughness· cwave taleing<•

I -4 ;l , e :;;= ;Q. ;I and T = 0 at M ~ .& plarte 11furL~ i!I: ~ (- J~ 2(- - -) and 4(-.-.-.). ,•

z ,....

/ \ ' / 0·6 \

i •, \ · ·. o-6 I \ \ I

\ : 0·4 I ,. I

· 0·2 I '-..·.,I

-0.02 ~: ·o-o) ·v {' 0 0 ····· " WJ.~ / \ I 0012 ,, ..

.17-0·2 \ /I I

· / l -0 4 I / ,. I I -o 6 I \ I ., -O·S I

• ..._ I ..... -1·0

[ 15

Fig. 5 : The transverse velocity profiles of a roughness wave taking l = I, e: = 0. I and T = 1 at X = 0 plane for n = I ( - ), 2(.:! - - -')and 4{-.~.~.); '

Fig. J5

zt 10

-- - I-~~, ,.... ,.... /

I ! I

\ l f..o 6 \ I I ' r0:4 ·; ' ',

' f'D2/ ' '{ ......

-0·02 -i 01 /--o I ·, 0 01

,~~-:::-'-::-;---~-3*~~~-.L.~~~..J...._~w1K . 0·02· / r\ / \ / -o 4 .

/ I ) ... \. :.

I -o 6 I \ . ' I . '.... ':.-6.ty- .

.............. ._/

-- ~:::-, 0

The transverse velocity . profi Jes of~ rou~h.~ess .. ~ave -~king "" l ...'... I, e = 0. _I and T = 2 at X . 0 plane for n = 1 ( - ), . 2(....,,-- -- -) a'.nd"4(;;1..;-.--.-;);

16 J

N1

,, \ . ,\ I \ I \

\ .WIK

0 i 0·011 0 02 ~0·03 IO·~'-·

I I

·!

~

: I i I I I I /

'~

I I

I I

I

( N1

/ ' . \ / \

/ \ i \

I ] ,\ • NI \

W/K ' , I W/K

.,!.," 'I I ,.,, J .~t 1,· ·~·012 ' o~ o·i;·,,~ I

I I . ' · · 1

\ i . . \ J

I \ \ \

\ '\ I ' ·I

@

I I

\

' ' ' • l

©

:;

I I

Fig. 7 The transverse velocity profiles ofa roughness wave t_aking

l = I, e: = 0. I at X = 0 plane for n = I (-), 2(-.-.-.-) and 4(- - - -) at la) T = 0, (b) T = I and (c) T = 2.

Acknowledgements

The authors.are thankful to the University Grants Commission,

New Delhi, for providing the financial assistance.

REFERENCES

[l] J. S. Citron, Slow viscous flow between rotating concentri~cylin

ders with axial roughness, J. Appl. Mech. 29 (1962), 188:...192.

. .

[2] Y N: Qaur, Slow unsteady flo-w of a viscous incompressible fluid

· between t~o w~vy.walls, Defence Sci./. 21 (1971), 1~12.

[ 17

[3] Y. N. Gaur and D. l\!I. Mehta, Slow unsteildy flow of a viscous

incompressible fluid between two coaxial circular cylinders

with axial roughness, Indian J. Pure Appl. Math. 12 (1981),

1160-ll70.

[4] H. K. Hepworth and \IV. Rice, Laminar flow between parallel

plates with arbitrary time varying pressure gradient and

arbitrary initial velocity, J. Appl. Mech. Trans. ASAJE 34

(1967), 215-216.

[5] H. K. Hepworth and W. Rice, Larninar two-dimensional flow in

conduits with_ arbitrary time varying pressure gradient,

J. Appl. Mech. Trans. AS,\IE 37 (1970), 861-864.

[6] D. M. Mehta and Y. N. Gaur, Time dependent slow flow through

a rough circular tube, J. A1ath. Phys. Sci. (Communicated).

[7] I. N. Sneddon, Fourier Transforms, McGrnw-Hill, l'\ew York.

1951.

[8] T. Sex!, Uber den von E. G. Richardson entdeckten Annular

-cffekt, Z. Phys: 61 (I 930), 349.

[9] S. Uchida, The pulsating viscous flow superposed on the steady

laminar motion or incompressible fluid in a circular pipe, z. Angew. Math. Phys. 7 (I 956), 403-422.

[IO] P. D. Verma, The pulsating viscous flow superposed on the

steady laminar motion of incompressible fluid between two

coaxial cylinders, Proc. Nat. Inst. Sci. India Part A 26 (1960)

447-458.

[II] P. D. Verma and Y. N. Gaur, Slow unsteady flow ofa viscous

incompressible fluid through a rough circular tube with

axial roughness, Indian]. Pure Appl. Math. I (1970), 492-501.

Jfianiibha, Vol. 15, I 985

RELATIONS BETWEEN THE HANKEL AND

MULTIVARIABLE H-FUNCT~ON OPERATORS

By

Y. S. Kumara Swamy and S. N. Mathur

Department of Mathematics, University of Jodhpur,

Jodhpur-342001, India

(Received: March 27, 1983; Revised: May 17, 1984)

ABSTRACT

The object of this paper is to obtain relations between the Hankel

operator and the multivariable H-function operator. Several results of

S.L.Kalla and R.K. Saxena[4],and of R.K. Saxena and R.K. Kumbhat

[5], become special cases of our main relations.

l . Introduction

The multivariable H-function due to H. M. Srivastava and

R. Panda [8, p. 130,Eq. ( l. l) J is defined and represented as follows

(See also Srivastava, Gupta and Goyal [7, p 251, Eq. (C. I)]:

fl : = H : [z:l] O, 11; 1111, Ill ; . ··; lnr, 11r [~l] (a;; a;<V ,. .. , OG/">1, p :

Z - p q·p q . ·p q. z (b·· P..(1) P..(1')) • r '' l· 11·~·' r, t r 3:tJ:1 , ••. ,t-"3 l1q•

(cJ<V, Yi) I, Pl; .•• ; (c;<•>, y;<r>)l, Pr J (d .(lJ "'.1) • • (d-<r> "'.(r))

j ' Oj 1 q .... ' J ' o, I q ' 1 . ' r

(2n~)T IL~· JL, <fo1(s1) ••• </>r(sr) 'P' (s1 ,. • ., Sr)

20 ]

7 S1 Sr d l • -I . . . Zr SJ • • · C. Sr (I.I)

where w =if-I,

m1 n; II r(d;<;> - i)(ils1) IT r(I -- c/il + y/'l s;)

,1.. ·( ) _ j= I j= 1 ~,, S·I -

~ p1 IT r(l - djU) + i)Ws;) II I'(c/il - y/ils,)

j=md-1 j =w+ 1

(i = l , .. ., r),

and

n r TI r(l - n; + ~ a;<ns;)

'Y(s1,. .. ,s,) j=l i=l

---~--" ---p I'

IT )=n+I

r r(aj - ~

i=l

q a/i>s;) IT

)=I r(l - b; +. ~ ~/ils;)

i=I

For further details and asymptotic expansions of the above function

(I. I), refer to [7] and P1J.

Banerji and Sethi [ l J have defined the multi variable fl-function

operator as follows:

a,~ Pc f(x) = ~\_-a-:;r,-1

(A,,) " . t' Jx [.\1 UJ 1s(x~-t~)af(t)H: dt 0 . ,\,, u

, .. ( 1.2)

a,~ i= s,.~ f(x) = x"' r<><-~13-1 u~ -- x~)a f(t) (itn) X

P1 VJ HJ : dt LAn V

···(1.3)

where

[ 2l

u (_t:)m; (i'- !;)n·1, x~ x'l!,

V = (·:_'l!,)nli (I - ~1:._)n; t ~ t '!!, '

and t;. n1i, ni are posilive numbers. These operators exist under the

following set of conditions:

(i) l .;;;;; p;, q1 < co, p;-1 + q;-1 =

r b (il

(ii) Re (3 + 1; ~ m; '-,-) > -1 ~;<· 1 q1

( ... ) R (" i: '~ b/i>) ) 111 e t' + .., "-' .n; c-- > -

l (:);"' q1

r h·w I (iv) Re (a + ~ J; m; -'-. ) > - -

l ~;\1) pi

{v) f(x) e: LP1

(0, co)

(J = 1, 2 , .. ., 111,.; i ,. . ., r)

The conditions (v) ensure that both P and Sexist. and also that

both belong to Lp; (0, co).

The operator of the Hankel transformation is given by .

(see Sneddon L6])

S' ... ,°' f = 2°'x-°' J~ y1-°' f(y) 12 ... +oi (xy)dy, · , .. (1.4)

22 J

where J,,(z) denotes the Bessel function defined by

I) ,. 2k+n oo <~~_!__z~-- _

J,,(z) = k~O 22k+n k! r(I-tn+k)

We also need [2,p. JO, Eq. (J7)]

Jx xm·:-n!.i+I r(m+l.) r(n+I)

tm-1 (xi; -tl.i)ri-1 dt = .. ~ .

0 rc~+l +n+l) ~

(Re ~ > O, Re n > 0, Re m :> 0)

and (3,p. 295, Eq. (3)]

Ioo 1,..-1 (tP - xP)v-1 dt = Xf'o+pv-p B(l - v -_!!:_, v), x p

where B(r1., {3) is the Beta function.

2. The Main Results

The following results will be established here :

.,p,, 'tl .. , 2 0, J1 + 2: ~.i, .n1 ; ••• ; mr, nr j[4. :.1 ]:: Pu: S 'l• 0t.f= x H . :

(,\,,) p+2, q+ I: P1, '11 ; ... ; p,,.,q,. /A1' ·

(a1; r1.;m , ... , «J{•>), ( c:~¥2k+2,YJ+2 . ~ J m1 , ••• , mr),

(b;; {3/11 , ... , {3;1'>), ( ~+~{3+2~+2k+2'1J+2 . ~ .

({3 + 2; n1 , ... , n,.):

~m1 + n1 , ... , m,. + Rr):

... (15)

••. ( 1.6)

... (I. 7)

'[ 23

(c/ll y·<llJ' · · (c·<r) y;<r)) ' ' 1,pl ,. .. , ' ' ·l.Jh..., (a.<v y/1>) · · (a.<r) y·<r)) .. J S',.,.0<. f

3 ' l ,ql ' ... ' ' ' 3 I ,qr .... (2.1)

ex, (3 ·o, n + 2 .. mu•n1 ; ... ; mr, n,. [41 I Sx· S'e, .f = x2$ H -.

(/\,.) p+2, q+ I: pi, qi ; ... ,· p,., q,. /\,.

(a ... N .n1 . ·a Jr.)) ( cx+~k+2.0+2e · m · "'1) (P.+1 · n n) · 1' .,,,,, , .••• , 'J ' ~ ' ··~1 , ••. ,,,.,, r,' t"" ,,),.1 , ... _,;_.r •

(b . r:i. n> P. (r)) { a+~+~(3-2k-20 . -+ + ) . ;, t'l , ... , t-l ' -. l~ . . . . • 'm1. ",n1., .. ~.,.f11r. nr .

( c .m y .<ll) • · ( c1<r) y .<r)) ] J o 3 I ,pl ) •• •) ) 3 I.pr I

(~·(1) y.<1>) • • (3.<r) y.lrl) S,(, i;f ' ' 3 l ,ql ' ... ' ' ' ' ·I ,qi . . ,,. (2.2)

Proof ~{:Z,l). Consider

a, (3 Ix [111u] I=. . Pf/) S' .,.,°'" f = x-s-1;'3-1. t8 (xii - tll)P H :. • dt (An) .• o . 1·~P

. J~ 2"' r"' y 1-"' f(y) 12.,,+°'" (ty)dy,

which, on replacing the fl-function by (l.l) .al'fd the Bessel function

by (1.5), becomes

I = 2°' ~x-s-i;P-1 J ~ yl-0< f (y) dy J: 1s---:et .(~e .,._ (t~pa

00 (-1 )k {ty)27ti;2<J+et

k~O 22k+,2'-l+~;k !,.F(l + .2;'l. +,,,a ¥~f J

.{~#'"'

24 ]

J ... r . fl cpi (Sil o/ (s1 , ... ,Sr) Li jL,

. (L) m; ~ s; (Ai)s' (I- ~- )ni·l: s; ds1 ... ds, dt. xii xi; ·

On changing the order of integration (which is justified under the

conditions stated with the definitions of the operators), we get

.. 1 = 2" i: -s-i; 2-1~ J 00

y1-" 1·c;· ;.) dy __ c,X ' (2n:w)'

0

J ... J Ilcp; (si} 'Y (s1 , ... ,Sr) Li Lr

('·)Si 1 d d ~ (-1)7' y27'+2'1+" • I\> -- Sl . Sr .:...

x~~ s; (mi+n;) k=O 221,+2·~+" k ! r(l +2·1J+~+k)

. I: f(3+2k+2'rJ+~ m; ~ s1+1)-l (x'.i-tl;) (~+IJ• 2: s;+l)-1 dt.

Evaluating the t-integral, using (1.6), we get

I = 2" ~ x-s-1;1l-1 Joo yl-,. f(y) dy . _1_ .. . 0 (2n:w)'

J ... J " Ilcp; (s•) '¥ (s1 ,. .. , Sr) L 1 Lr

00 ...

,\; Si . • } . ~ (-1 )h: y2H2'1+0t ' ( ) x~ 2:s,(m,+n;) k=O 22k+2'1+°' kl r(l +21J+a+k)

L 25

• ~-1 xa+2k+2'1J+~ ini 2; st+ 1 +~~+~n; ~ s;-f- 1+1

I'( 1>+2k+2·1)+\m•Si+l+I )r(~+'1Jil:s1+I+l)

I'( 8+2k+2·1J+~ 111i ,~ Si+l + l +~~-\-~ni 2; Si+~+~ ds1 dsr. . ~ )

0, n + 2: mi, n1 ; ... ; mr, n, [''1 I = x 2 H . p-f-2, q+ I: P1, q1 ; ... ; p,., qr . ,.\,.

( (1) (r)) ( 8+2k+2·1)+2 . ) (~+2· . )· a;;11.j , •• .,aj' -~ -.1111;.,mr, ,n1.····llr·

(bi; ~/ll , ••. , ~;<r>), { 8+~~+2~+2k+2'1J+ 2 ; 1111 +n1 , ... , mr+n,):

(c .m · .;.<r>) • • (c·<>·J v.(rl) ·. ' ' ' ' I ,pl ' .. ·' i ' " I ,pr J

(a.m y·m) . · (a.<r> y.<r>) 1 ' ' l ,ql ' ... ' ' ' ' 1,q,

. 2• .xc• I~ y•-• JM• (xy)dy

This completes the proof of (2. 1).

Similarly, by using (I. 7), we get the result (2.2).

3. SPECIAL CASES

Setting n=2 in (2.1) and (2.2) we obtain the relationships between the opera wrs studie<l in [5]. The results of (4 J can be also obtained by further speciaiizing our results.

Acknow Iedgemeuts

Our thanks are due to Dr. R. K. Saxena, Professor of Mathematics, University of Jodhpur, and Dr. B. L. Mathur, Defence Laboratory,

Jodhpur, for giving all help. We are also grateful to Professor

26 ]

H. M. Srivastava, University of Victoria, Canada, for giving sugge

stions for revising of the paper.

REFERENCES

(1] P. K. Banerji and P.L. Sethi, Operators of a generalized function

of n variables, Math. Student 45 (1978), 152-159.

[2] A. Erdelyi et al., Higher Transcendental Functions, Vol. I, McGraw

-Hill, New York, 1953.

[3] I. S. Gradshteyn and I. W. Ryzhik, Tables of Integrals, Series

and Products, Academic press, New York and London, 1965.

[4] S. L. Kalla and R. K, Saxena, Relations between Hankel and

hypergeometric function operators, Univ. Nae. Tucuman Rev. Ser.

A 21 (1971), 231-234.

[5] R. K. Saxena and R. K. Kumbhat, Fractional integration oper

ators of two variables, Proc. Indian Acad. S<:i. Sect. A 76 .(1973),

177-186.

[6] I. N. Sneddon, Fractional Integration and Dual Integral Equations,

North Carolina, 196'2.

[7] H. M. Srivstava, R. C. Gupta and S. P. Goyal, The H-Functfons

of One and Two Variables with Applications, South Asian Publi

shers, New Delhi and Madras, 1982.

[8] H. M. Srivastava and R. Panda, Expansion theorems for the

H-function of several complex variables, J. Reine Angew. Math.

288 ( 1976)' 129-145.

•

Jfianabha, Vol. 15, 1985

FIXED POINT THEOREM FOR MAPPING WITH A

GENERALIZED CONTRACTIVE ITERATE IN

2-METRIC SPACES

By

V. H. Badshah and Bijendra Singh

School of Studies in Mathematics, Vikram University,

Ujjain-456010. M. P., India

(Received: July 22, 1983; Revised : June 1, 1984)

The concept of 2 metric spaces has been investigated by S. Gahler

in a series of papers ([3] to [6]) and other related works have been

done by himself and others for extensive bibliography, we mention the

work ofK. Iseki [8].

The notion of contraction type mapping and fixed point theorem

in 2-metric spaces has been introduced by K. Iseki [7j. Da~s and

Gupta (I] generalized Banach contraction principle for mapping of T

satisfying

d(T,,, Tu) ~ex d(y, Ty)[! +d(x, Tx)l + ~d(x, y) [I +d (x, y)]

for all x, y e X, ex, ~ > 0, ex + ~ < I.

The object of this paper is to establish a fixed point theorem for

functions satisfying generalized contractive type condition which

generalized the result of Guseman [2].

Lemma. Let X be 2-metric space, and T : X _,,. X be a mapping.

Let Bex with T(B)CB. If there exist a UfiB and positive integer n(u)

28]

such that Tn<ul(u) = u, and (for some non-negative constants a, ~ with

ix+~<:. 1) let

(I) d(Tn(•ll(x), Tn<•tl(u), a)~ a d(u, y,.("'(u); a) [l+d (x, Tn<i•l(x), a)l [1 +d (x, u, a)]

+ ~ d (x, u, a)

for all x, a E X. Then x is the unique fixed point of Tin B, and I . . .

Tn(yo) -> u for each )'oEB.

Proof By (1 ), u is the unique fixed point of T"<"J in B then

T(u) = T (P< 11 J (u) ) = Tn(ul(T (u) ) => Tu= u. But u is the

unique fixed point of Tin B. Let J'oEB, and note that

T(B) CB=> {P(yo); /1 > l} C8.

Let r (y0 ) = max {d (T"(yo), u, a); I ~ m < n (u) -1} and for

all n sufficiently large with n=r. n (u) +s, where n (u) =p, n=rp + s,

with r > 0, 0 ~ s ~ p - l, ~uch that P(u)= (u).

Then

d(Tn(yo), u, a) ~ d(T"(v0 ), u, P(u)) + d(T"(Yo), TP(u), a)

+ d(TP(u), u, a).

T ..i::: is can be written as

d(Tn(yq), u, a) = d(T"(Yo), TP(u), a)

== d(TriH--<(y0), TP(u), a)

= d(TP T<r-l>P+•(yo), Tt'(u), u)

d(u, T"(u), a) [I_+d (T<i-lliH-• (y0 ),

['%9

TTP+s (yo), a) r ~- rt [1It*'·d11f(T< r"'ll!11+ •(yit)Y;Uj'U} J

+ (j'd('ffr4~v¥i(y11), u, a)

= I' d(T<r-I>P+•(yo, u, a) < 1'2 d(T11-I>P+•(yr, u, a) < ...

< W d(T• (y0, u, a)

< {'7 r(yo)

and it follows that Tn (yo) ~ u is finite .

. Theorem. Let (X, d) be a comlete 2-metric space and let T be a

mapping of x into itself. Suj>pbse there exists a sex; such that

(a) T (B)CB.

(b) a + I'. < 1 and ea.ch y " B; tbire is an integer n(y), > 1, with

d(Tnl11> (x), Tn<111 (y), 4 ~-- adfy, :J7n<'f~;€.f), a}fl +d(x, y11<11 (x), a)] [I + d(x, y, a) ]--- --

+ ~ d(x, p, a)

for all x E B.:

(c) For some xo"B, Cl {'f~xb); n ;$ l} CB.

Then there is a unique u '" 0/mch thtttT(u) = u Punherinore, if

d(TnWlt~), '']Mfi111~,.iz;)~· cx..d(y,Tn<11>(y);.B.~J'~·+.,d{x,,,,T11<v1 (.,),a)] [I+ d(x,y, a)]

+ I' 1d(x, y, ·a'

,for all a. x "X, then y is the uniquefixedpointin X, a11d Tn (xo) ~ u

for each xo " X.

[' 31

Similarly,

d(x3, x2 , a) = d(Tm1 (Tm~(x1 ), xi, a)

. . m .· ~ ~ d(T 2(x1), (x1), a)

~ ~~ d(Tm2(xo), xo, a).

Proceeding in this way we can get

d(x1+h Xi, a) ~ Wd(Tm1(xo), xc, a), i ;;;;,,

~ ~tr(xo) ·,i~l.

By routine calculatron one can sh<)w that the following inequality holds

j-1 d(x•, x;, n) ~ ~ d(x1.,.i. X1, a), j > i ·

l=I

<' ()i ( ) . . -.... -- r xo , ; > 1. 1-~ ••

. ' Itfollows thatthe sequence {x1}.is a Cauchy, using:completeness

and (c), we have Xi-+ u € B. There is an integer n(u) ~ 1, such that

d(Tn<ul(y), Tn<ul(u), a)~ a d(u, Tn<ul(u), ~1 + d(y, Tn(U)(y), a) J . [I + d (y, u, a)] --

+ ~d(y, u, a)

for each y € B.

It implies that Tnlul(x1) -+ Tn(u).

32'' 1

Then d(T11(u), a) = lim d(T11(x;), u, a). i-H>O

However,

d(T11(x;), Xi, a) = d(Tn Tmi:_1(XL1), r11'-1(x1_1), a)

~ ~d(Tn(fL1), XLh a}·fTom (A)

~ ~id(Tn(x0), (x0), a) ~ 0 as i ~ oo.

Hence T11(u) = u. By the lemma, u is the unique fixed point of

Tin B, and T"(Yo) ~ u for cadi:\yo E U. The last assertion of the theorem follows directly from the lemnia.

REFERtf~\~ . .

[I] B. K. Dass and S. Gupta, An extension of Banl;lch contraetion

prfociple through rational expression, Indian J. Pure

Appl Math. 6 (1975), 1455-1458~·

(2] L. F. Guseman, Jr., Fixed point theorems for mappings with a

cdiftr~i\l~'t¥et§'&' a°f k pb1tif, Pfo't;t; 1#i"&r'. t/&:1~. s'd<:. 26 (1977), 615-618.

[3] S. Gabler, 2-metriche Raume and ihre fopofogisohe struktur,

Math. Nachr. 26 (l963/64), 115-:.i'ilk. [4] S. Gahler, Linear 2-normierte Raume, Math. Nachr. 28 (1965),

1-43.

[5] S. Gabler, Uber die unifor~isier ba~keli 2Smetriche Raume,

Math. Nachr. 28 (1965), 335-244 .. t~ri ~: ~~r Ztut &C'&'me't:fic' 2~1ii~~ Tf.~~.· 11f,,: 11li>umaine

Mflfh; Pi:ttes Af!pl: +1:· {f96'6), 66~~~;

(7] K. lseki, Fixed point theorem in 2-metric spaces, Math. Sem.

Nt1te1i&{j(f· f!ftfv. ·j cif'975) 13~"-1~6'1 [8] K. Iseki; ·~atn8'fiiati'cs 1in 2-normed spaces, Math. Sem. Notes

KObe Univ 4 (197'6), ·161~1'74';

[9] S. L. Singh, Sctme contractive type principles on 2-metric spaces

and applications, Math. Sem. Notes Ko/Je' Univ. 7

(1979) 1-11.


A FIXED POINT THEOREM .FOR L-CONTRACTIONS, I~

GENERALIZED METRIC SPACES.

By

A. CARBONE

Universita degli studi della Calabria, Dipartimento di Matematica, ··

87036 ARCAVACATA di RENDE (Cosenza) l'!A~Y

' • (Received :. A~gust6, 1984) :}

ABSTRACT

The aim of this note is to extend a result concerning the .. existeµce

of fixed-point for contractive mappings in generalized metric spaces to

. a more genera/ dass of mappings .. . ,

Befo're stating our result, we introduce, for sake of co~pJ~t~~ess'; · · some &tandard noti011s [l"].

:!'

Let Ebe a Banach space. A subset K of Eis called a cone if it is

closed, convex, tK C K for t " R+ and Kn (-K) ...:... 0. ,.;

Given a cone Kin E we_define ap_artlal ordering in Eby writing

x < y if and only if y - x " K.

Moreover, K is.called normal if there exists a.> 0 such that·

0 < x ~ y implies 11x11 ~a 11 y II .

. We shall assume in the fo Bowing that K is a normal cone.

Definition. A topokigical spac.;: Xis said to be a generalized metric

space if there exists a function d: Xx X ~ K such that: '··

(i) d(x, y) = 0 " K <o>. x = y; · .;

34]

(ii) d(x, y) = d(y, x);

(fof'·dt~t, Y) ·z. d(x, z) + il(Z, y)

(" <" denotes the partial ordering induced by K),

Let/: K - R+ be a sublinear positively homogenous functional

(that is, if u, v E K~ then

l(u + v) ~ !{ti) + /( v) arid l(/u) ~ t f(uj for i $ 0) such. tLat

1-1(0) = 0. Then; if d*: X x X .:.+ R+ iS Hie f m~ction defined by

d*(x, Y) == l(d(.X, · y) ), (1)

w4'·lif:il!"e thaf(x, d ')is a:· metric sp'ace.

We s.hall say that a generalized' metric sp'ace i1S i to'ii/}>ntl g'~ne ralized metric. space ifit is complete in the rr.etric defined by (1 ).

· :&f-a.~cti'inplbte- generalized. ir.clric space (x, d) the following result

was rroved in [1, Theorem 6. 2].

let T: x -.x be;a• rrjap stich\tha't

.d('T(x); T(y)) <·Ed('x, J ); (2)

where L is a positive (L(K) C K) and bounded linear operator in E

with spectral' radius;. r(L )i le'ss tha:lf I~ If2 x is cY:impMtS~'. gerteraiized

metric space with l(u) = II u 11n, tha11 F h'as a un'it[HcsrixeCf poiht wJ11c:1

is the limit of the successive approximations

x,.+1 = Tx,. n = 0, 1, 2

for any initial va-l<uetx:' =" ·x0•1E X:~ .

In the Theorem below we shall show thahf T: x- X·is not a

[ 35

J ·/' -, \ <". ".. "-·~ f i ./ . f

· necessarily contin'uous map which satisfies a relaxed condition than (2)

then the same result holds.

More precisely we have the fo!Lnving

Then.rem.Let X be a generaliz,ed metrir: space which. is com.(J(:ete in the

metric (1) with l (u) ~ II u ii ( 11 u fl s~~~ds f~r th~ r.or; in E).

If T: -> X satisfies

d(T(x), T(y)) < L[ d(T(>.), ,\) + d(T(y), y) ], (3)

'; .;,. ,:'· .' . ·: ;.l" +1 tj

where L is a bounded positive linear operator in E with spectral radius

smaller than t, then the equation

x = Tx

has a unique solution in x, which is the limit of tke successive approximations

x,..,.1 = Txn, n = 0, I; 2

for any initial approximalion x' "X.

Proof. Since L is sublinear, it follows from (3) fo~ x ~· Tx' and

y = x'

d(T(Tx'), T(x')) < Ld(T2(x'), T(x')) + Ld(T(x'), x').

That is

(/ - L) d(T2(x'), T(x')) < Ld(T(x'), x ).

Being r(L) < t < 1, ""·e have that (I - L) is invertible, hence

.• _,,.

d(T2(x'), T(x')) <(I- Lr1 Ld(Tx', x'). (4)

Clearly we have that

36]

d(Tn+l(x'), Tn(x')) < (l - L)-n Ln d(T(x'), x').

Indeed, for n = l, it has been already proved.

Assume that (5) is true for n = 1, then

· ·d(Tn+l(x'), Tn(x')) = d(T(Tn(x'), T(Tn-l(x')) <

< Ld(Tn+l(x'), T 11(x')) + Ld(T"(x'), Tn-J(x')).

That is

(T- L) d(T~+l(x'),Tn(x')) < Ld(Tn(>. '), Tn'-l(x)).

By the inductive assumption

(5)

(l - L) d(Tn+l(x'), Tri(x')) < L(I - L)-<n-1i Ln-1 d(T(x'), x').

Since

L(l - Lr<n-u =(I_ Lpn-v I,

we obtain

., (l.-L)·cl(Tn+l(x')• Tn(x')) <(I-,-L)-<n._VLnd(T(x'), x').

Thus (5) is proved~·

Furthermore; we have

cl( Tn +m(x '), T 71(x '))<cl( Tn+m(x '), Tn+m-J(x') + d( Tn+m-J(x'), Tn+m-2(x ')) +

+ ... + d(Tn+l(x'), Tn(x')) <

< {(I _ L )-<n+m-ll Ln+m-1 + (I - L )-in·l 1ri-2l £n+m-2 + ... +

+ (I - L)-" Ln} cl(T(x'), x') <

[ 37

00

<.<.I-LP'//• ( ~ (I-Lrm-u Lm~1 (q(T(x'), ~'))) :="'. (l-,-LtU~)n u' .. .. m·=J ...

where u' is the unique solution of

u = (I-L)-lLu + d(T(x'), x'_).

Indeed, from the spectral. mapping theorem, we have that the

spectral operator.{1-L)-lL is such thaf r((I-L)-1 LY< l.

Hence

d(Tn+m(x'), P•(x')) < ((I:_Lt1L)11 u'

Being K normal, we have :--.

II d(Tn+m(x'), T 11(x') II ~ cr II ((I-L)-1 L)n u' II.~. {6)

Since the right-hand side of (6) is going to zero ~heipr-,+ + ?"•

we obtain that {T1'(x')} is a Cauchy-sequence with respect to the metric

d*. Being (X, d*) complete, we denote by x* the lemit of {T11(x'.)}.The11

the following inequalitieshold

d(T(x*), x*) < d(T(x*), T 11(x') + d(T11(x'), x*)<;

. :·~ (i-L)-1 Ld(x*, rn-1 (x')) + d(T11(x\ x*):

Finally, using the normality of K,

11 d(T(x*), x*) 11 ~ cr ( 11 (l-L)-lL 11 • ii d(x*.• T11-I(x') 11 +

+ 11 d(T11(x'), x*) 11 ).

Letting n --+ + oo in (7), we obtain

T(x*) = x.*.

(7)

,.

;':,

38]

At last, we would like to add in passing that the Theorem would

~be still ttue if the :riiap Tis a "gen~falized codtraction"· tna.p (cfr.

[2 Ch. 1] for an extensive bibliography).

'··' REFERENCES

[lJ Krasnosel' skii M.A., Vainikko G. M., Zabreiko P. P., Rutitskii

: Ya. B.;;Stetsenko V. Ya., A.pprdximate sOlut'io:~r /Jf"ope/-ator equa

tions._ Woltet-'Nordhoff Publishing, _Groningen 1972 ...

[2] Rus I. A., Metrical Fixed Point Tn.eorems, University of Cluj

Napoca, Departme!1t of Mathematics, Cluj-Napoca 1979.

"'

(1 ~rt

·)

Jfianabha, Vol 15, 1985

,· IN;IEG~~J..S J1'J:VOLNlNG . . -t\ .. GE;NE!t;AL CLASS PF

.PcOLY;NOMJ~,LS

By

A; K Arora ~nd C. i! :Kdul

tlepaftment of Matb;_..:Ua~ics, M. R. E~gineering C;_llege,

Jaipur-302017, Rajasthan, India

tRe,r;e~y~d : A µgust, .17,11984)

ABSTRACT

JI,,M. ,Srivas~ava a,n;d. N:;P.,Sipgh [4-J _recenpy,. established

integrals involving certain products of a general class,,~f .poly,noJ.nials : .- ~. ·• ·. " ' .~ ·"' ; ... . ' ;..; ·:..

S1/ 11 (x) and the multivariable Fl-function. - The aini of this paper is

to fm ther extend these results and estatlish certain .integrals «J:>oth

si1:gle and multipk) involving the products of S,. m. (x) a~Jtiie. m~lti-varible Fl-functions. ·, ~~

1. ~N-TRODUCTioN·

. / / Srivastava [ 2, p. 1, eqn. ( l) ] introduced a general class of poly-

no1mial~:-?'1i"' ~~).f:iefined by. means of the following e,quation-

(1.1) sm [n/m]

n [xJ = ~. (-n),,,k k=O k!

A11,k-X1',-rn = -0, l, :2,< '~··"

; .._: , ; ~ ! i

where m is an arbitrary positive integer and t.he. coefficients • ' .. "•· ·;;;<!

An, k (n, k > 0) are arbitrary constants.

On specializing t}J.ese arbitrary cdefficients A,., k, Sn'"'i (r1}rY.i~lds a

number of knowri.polynomiitls as special cases. Thesdriclude, among

40]

othors, the Jacobi polynomials, the Hermite polynontials, the Laguerre

polynomials, the· Gould-Hopper polynomials, Brafman polynomials,

etc. ( see ( 4 J ). Srivastava and Singh [ 4 J recently established in

tegrals involving the products of S,-.m (x) and certain multivariab1e

fl-functions. In this paper we further extend these results and esta

blish integrals (both, single and multiple) involving the products of

Snm (x) and muhivariable fl-functions. Our results are thus capable

of yielding a number of integrals involving products of various poly

Mmials with the multivariable H functions and other special functions

to v.hich the multivariable fl-functions reduce.

ln the se4 uel, we also require the following general class of

polynomials of r variables :

( 2.) R··mt> .. .,mr·c· ) ·l. Xi. ... ,x, n .

J~n r k E .. (-n)JA(n;ki.~·-,kr)_TI {xi 'Jki!},

k]> ... ,kr = 0 l= 1

'I

where J =;= E (im •. k.), m-1 > I, i = I,. .. ,r, and the coefficients i=l .

A(n; kJ, .. .,k,),.are arbitrary constants.

The multi variable H-f unction is represented here in the following

manner (see ( 3, p. 2.51, eqn. ( c. 1 ) ] ) :

lz1 J 0, N: Ali, N1; ... ;M,, Nr [z1 , . (l.3) ;H I . : = H . . : .

. LZr , • P, Q: P1, Qi; ... ;P,, Qr Zr

'\

{ 41

(a .. N, N <1;) . (c'· y'·) . · (c-<r> 1 .<rl) '' ~ i, ... ,.,,; l,P · '' ' 1,P1 , ... , ' ' ' l,P, l

(b .. fJ.I A .(rl) • (d . '>' ·) • • (d·(r) '> .(rl) J' J' " i>- .. "'' . l Q . ,, 1) 1 1 . Q . ' ... ' . ' . 0.1 . 1 0

' ' 1 '..._,r

For the definition of Hl ?] and the various conditions on the para-L z~ - .

meters as also the conditions of convergence, etc., we refer to

[ 3, p. 252, eqns. (C 1) to (C. 8) ]. These conditions are assumed to

be satisEed throughout this paper. Also, the appearance of an asterisk

( * ) at a particular place indicates that parameters at that place are

the same as on tne right side of (1.3) at the corresponding place,

2. THE SINGLE INTEGRALS

For i = 1 , ... , rand with

X;(x) = (x-CJ.li (p-x)q;, p;, q; > 0

Yi(x) = (x-y)-pi-q; X1(x)

and

f(x) = (x_.:CJ.yu-1 (p-x},-1 (x-yfu-v

g(x) = (x-a)P. (~-x)A (x-yy-p.-A, µ,, ,\ :;> 0 (both not zero

simultaneously), we have :

JP · [z1Xix)]

(2.1) . (x-a)u-1 (p-x)"'-1 s; [a(x-a)I'- (p-X)A] H ; .. dx OI: Zr X,(x)

= (p-a)u+~-1 ~ -n)~k A,,, k a1' (p-o.)7•<1'"+)..> [n/m]{( k=O k.

42 1

0 N+ 2 . * . . * [ p +q I E . * * ]} , . , · · ·, (B ) 1 1 · , · · ·, . . ~ -a

• H P+2, Q+ I : * ; ... ; * (> )P,+q, F: * ,. .. , * Zr ;-J-:i

J~ . f z1 Y1(xr1

(2.2) ·· f(x) s; [ g(x) ] H ; . J dx IX ~Zr Y,(x)

= R :S ...::!!....:!!:' An,A [a{~-a)l'-U. {~-yp• {a-y)-J.}k [n/mJ f ( . ) . . k=O Id .

. H ' ,, . . '..... . ( : 1 .. , '· ,. . ., . , y < a < ~ 0 "'+2 · * · · * [Z E · * · · *]} P+2, Q+I. , .. ., .. Lz. / F. ,. .. , .. . . . * . * . . *

with

R = (~-a)u+u-1 (~-yru {n-yp

and

p; I q; -pi -qi Z1 = Z; (~-oc) 1 (~-Y) (a-y) , i = I , .. ., r ;

(2.3) J~ f(x)S;[(-·I)""+Aag(x)JH[(-l)PI+qi71 Y1(x) ]dx a (-Il'+q, ;, Y,(x)

_ R :S . -n mk An,k [ a(~-a)""+A (y-~f""" {y-u):-J.jk [n/m] f ( ) k=O k! .

, iv • , ... , (-1) I _1z1 , . , ... ,

. Fl : ' 0 "+2 · * · · * [ p +q· l E · * · · * 1}

P+2, Q+ 1 : * ; ... ; * (-l)Pr+qr z,. F: * ; ... ; * J

a < ~ < y, where Rand Zi ar.:! defined with (2.2) above.

[ 43

In equations (2.1) to (2.3),

E: (1-u-µk; P1 , .. ., p .. ), (1-v-,\k; qi,. . ., q,), (a;; a.';,. . ., a/r>)l>Q

F: (1-u-v-(µ-t A) k; Pl +q1 ,. .. , Pr+qr ), (b;; ~/ ,. . ., ~;<r>)i.Q

and (2.1)-(2.3) are valid if

r r min {Re ( u + ~ pt D/i>, v + ~ qi D/i>) } > 0

i= l i= 1

where

(2 4) D/i> = d/il/3;<il, j = I , .. ., M., i =I , .. ., r.

3. THE MULTIPLE INTEGRALS

For i = I ,. . ., r and with

U; (xi) = (x.-;.;)P 1 (~; - x;)q 1, p;, qi > 0

W; (xi)= (xt-Yi)-pi-qt Ut (x1)

µ- · Ai V;(x1) = (ri-11.i) ' (~1-Xt) , µ;, t\1 ;;i. 0 (not all zero simultane-

ously),

( ( -µi~t\; ( Ti Xt) = Xi-Yi) v. Xi),

· '!:... and

fi'I ) ( Ui-1 ( V;-1 ( -Ui-Vt \Xt = Xi-11.i) ~· - Xi) X;-yi) ,

we have:

44]

J ~1 1~· r u·-1 v·-1 (3.1) ... . II [ (x1-a,) ' (p;-X;) ' J X

°'I arz=l

. R':/1 , •.. ,m, (a1 Vi(x;) , ... ,a, V,(xr)) H : dx1 ... dx, [

ZI U1 (x1)] ·

Zr Ur (x,)

r . J~ n = . II (B1) 1: . { (-n)J A(n; ki, ... ,kr)

I= J ki,•·•,k,.=cO

k· ( + ~ r~ '(p;-a;) µ; /\1)k; )

i= 1 kt ! 1 0, N: M1. N1+2; ... ;M,, N,+ 2 [ z1(~1-11l1 +qil *: C]l

. HP, Q: Pi+ 2, Qi-1-:::. l ; ... ;Pr+2, Q,+1 . Zr =(~r-1Xr)Pr+qr * : D )

\Vith.

B ( u1+v,-l . ; = ~;-a,) , 1 = I, ... ,r;

J P1 · J ~r r

(3.2) ... .n. [ f(x;)] R:1, •.. ,m, (a{T1(x1), ... ,a, Tr(Xr)) a1 IXr l=I

JZ1 W1 (:X1) J . H ! . · dx1 ... dx,

LZr Wr (x,)

r J ~ n . r [ kt h-k;, J . 2; (l'Ji) II {(-n)J A(n;k1,. •. ,k,) _IT a; . , • x 1=1 ki,···,kr=O Z=J k, •

0, N: M1, N1+2; ... ;M,, N,+2

[~/I *: Cll •HP, Q: P1+2, Q1+1; ... ;Pr+2, Qr+l Zr') *: D Jl

for y; < a; < ~; , i = I, ... ,r, witft

u;+ Vi-- l ( -Ui ( -V.: ·!Ji = (~i-a;) ~•-Yi) ()(,·-y;)

µJ+A, ( . ~µ; ( -A1 h« = (~;-()(i) ~i-.:y,) r.;;-yi)

I (rJ. )Pi+qi ((). )-p;, ( )-qt • 1 Z;, = z; ";-a; t-'i-Yi «;-y; , l= , .. .,r;

(3.3) J ~l I~' r ... 11 a1 · «r .i=l

[f(x;)J

Rm1,. .. ,mr ( (-1)µ1 +Ai aiT1(x1), ... ,(-l)µ.r+Ar arTr(xr)) n

H[(-1) Pr ·Hi ? W1 (x1) J Jx1

... dx,

(-l)Pr+q, Zr Wr (xr)

[ 45

r J.,,;;;n . r [(-l)µ.;+A; a:k;h;k'] = TI {-1ji) 2.:: { (-n)1 A(n, ki, ... ,k,) Il - - ----·- 1 k k -0 . ·-1 k; ! l- · 1, .. ., r- l-

0, N: 1i.f1, Ni +2; ... ;M,., N,+2 [ (-l)P1+q1 Z1' I * : CJ} . H P;Q: P1+2, Q1+1; ... ;P,+2, Q,+l (-l/'+q, t,' *: D ·

for a; < ~; < y;, i = l_, ... , r,

'Ahere "!Ji, h; and Z,' are given above with (3.2).

In equati'Ons (3.1) to (3.3), we.have

C: (l-u1 - µ.1k1,P1), (l-v1-1\1k1' qi), (c;', y/)1 p ; ... ; ' 1

(1-u.,-µr kr, p;-), (1-·v,--A,k.r, qr), (c/•>, y/•>)1,Pr

D: (d/, 3; )1 Q '(l-u1-V1-(µ.1+Ar)ki.P1+q1) ;: .. ; , 1

46 ]

(d/1\ 3/")l ,Q, (1-ztr-vr-(µr+,\, )k,, p,+q,)

and the conditions of validity of (3. I) to (3.3) are:

min {Re (Ui+p; D/1', v1+q; D/ii)} > 0, j = 2, ... ,M;; i=l, .. .,r,

where Dt:1 bei11g given by (2.4);

l f"1J1+woo I"IJr+woo ( r ) ·-a; (3.4) --- ... exp L; s; t; [(t1+p1) ]

( 2rtw)' · 1 J '1)1-woo "ljr--woo z=

Rm1,···,mr ( ( -,\) -,\ r z (t +P )-/l·l I · n 01 t1 +P1) , .. ., a,(t,+ Jl1·) ' ) H i 1 :1 1 ,dt1 .• dt,

L Zr(t +P f P-' l .

r a.-l r J<n = . TI (s; ) exp (- . 2; s; P•) L; {(-n)J A(n; !q,. .. ,k,)

z-=0 z= 1 k1,. .. ,k,=O

,. r /{i luk i] TI I a; s;

i= I L k, t

0, M1 , N1 ; ... ;M.-,N, [z1 ?µ~I : : * ; ... ; *J} Zr Srµ' ·Di

.H P, Q: P1, Q1 +1; ... ;P,, Qr+ 1

(w = if--1) with

D • (d·' "'·') (1-H -\ k µ )" ·(d·(r) '>.(r) '>.(r)) l • 1 ' Oj l Q ' wl . 1 l• 1 '•••, J ' 03 ' 03 } Q ' 1 , r

(1-a,-Ar k,., µ•)

provided that,\;, µ1, s; > 0, 'IJi +Re (p1) > (J, Re (a;) > 0,

r Re (ix;) + ·r;; l,' µ; > 0, 1 = i, . .,r;

j=l

( 47

1 J'f,1 +woo J'tJr+woo ( r ) r (3.5) 271:w ,- ... . exp . ~ s; t; . TI ((t1+p;)-1Xi ] ( ) 'IJ1-w00 't]r-woo l = 1 l = 1

· R~11 ,. .. ,m,. (a1(t1 +P1)-,\1 ,. .. , a,.(tr+p,.f,\,. ) H[zi(~i +Pt)µ.1 Jdt1 ... dtr

Zr( tr+ pr)/.k''

r r a;-l J<,,n = exp (- .z: Sip.). n (s; ) z; {(-n)J A(n; k1, ... ,kr)

l= 1 I= 1 kl,. . .,kr=O ·

r TI

i= 1 {

. tu . ,\; k;} 0, N: Afl' Ni; ... ;M,., N,. (~--- H

ki I p , .. p I 1 Q p + 1 Q • ., '::f • . l T ' 1, ... , 'f ' I'

rz. s1·-µ, i *: C1 l}

L I

L.-- : -µ II * : * ; ... ; «J .:,,· Sr

(w = v-1) with

C1 : (Ci', y/)l,Pi, (rx1 +>.1 k1, /k1) ; ... ; (c/r>, Y;('" 1)l,P,. ,. (1Xr+>.r.h, µ.r)

proviLL'd that ,\;, µ; > 0, ·IJ: + Re (pi) > 0, Re (u;) > 0,

r and R.c (ix;) > ·;i; · :.>..; µ; , i = ,. .. , r.

j=l

Method of proof : To establish (2 1), we substitute for Snm [xj

occurring in the integrand on the left of (2.1 ), from ( 1.1 ), and then

integrate term-by-term to get

~/m] (-n);"k An,i.: akJa (x-a)u+µ.k-1 (~-x)v+>.k-1 k=O k. ~

~·~· J

.. ·rzl xl (x) J fl : ' dx. Jr X1· (x)

We next substitute for the multivariable ff-function occurring in the

it~tegrarid of the above integral foterms of 'the m~itiple contour . . integral from [ 3, p. 251, eq11. ( C. 1) ], change the order of integra-

tions, evaluate the x-integral ·by means of [ 1, p. 10, eqn. (13)] and

tl:en'frit~rpret the res\!hiJ'.g integral with 'the help of [ 3, p. 251.. eqn.

( C. 1) ] to get the desired res.ult.

Formulas (2.2) arld (2.3) are established similarly with the help

of [ 1, p. 10, eqns. (14),·(f.) J.

The multiple integrals (3.1 )'to (3.3) are established with the help

Of (1.2) in a similar manner, while as the multiple contour integrals

(3.4) to (3 .5) are established with the help of the following result :

--- e'" 1 (t+ /!)-" dt = s ____ e_ , s > 0, l 111 twoo C<-J -sp

2rcw -r,-woo f((I()

(w = .V-1), where, for convergence '1J +Re (p) > 0 and Re (a)> 0.

4. Particular Cases

(i) If, in (2.1 ), we take fh =0 and .\= 1, it reduces to the known result

[4, p. 166, eqn. (2.2) ].

(ii) 'If, in(3.1), we replace the general class of polynomials of r

variables, viz.

R~1 " .. ,mr (x1,. .. ,xr) by the product of r polynomials of the type

s;:i (x;) it yields :

[ 49

J ~l I~r r { Ui-1 v;-l m1 · } (4.1) ... . TI (x;-a;) (~;-x;) S ni [lli Vi(x1)J X a 1 ar l=I

[

z1 U1 (x1) J • H : dx1 ••• dxr

Zr Ur (xr)

_ r [n.ifm1J [nr/mr]{ r [(-n;)m;k; A a;k1 (~i-a;)(µ1+A1)k;l - TI (8;) 1: ... ~ TI n· k· I"

0 . 1 ,, • J i= 1 k;=O kr= 1= k1 !

O,N:M1,N1+2; ... ;Mr,N,+2 [ (~ )P1+q1r*:Cjl .HP, Q: Pd-2, Qi+l; ... ;Pr-+ 2, Qr+ 1 .:: (~=:lr+qr *: D J

where U,(x;), V.(x;) and B;, C, Dare given above. The conditions of validity of (4.1) are:

min {Re (u; + p; D/i>, v; + q. D;< 1 i )} > 0, j=l , ... ,Mi; i=l , ... , r, v.here D/il being given by (2.4).

ACKNOWLEDGEMENTS

The authors are grateful to Prof. H. M. Srivastava for his kind encouragement during the preparation of this peper. The authors are also tha,1kful to the U. G. C. for providing financial assistance.

I

REFEREN.CES

( I ] A. Erd1;·Jyi at al., Higher Transcendental Functions, Vol. 1, McGra.v-.liill, New York, 1953.

[ 2] H. M. Sri>a.stava, A contour integral involving Fox's Ii-function, f,;dian J. Math. 14 (1972), 1-6.

[ 3 ] H. M. Srnastava, K. C. Gupta and S. P. Goyal, The fl-Functions of One and Tv.·o Variables with Applications, South Asian Publishers, New Delhi and Madras, 1982.

[ 4] H. M. Sriva~tava and N. P. Singh, The integration of certain prcducts of the multivariable Ji-function with a general class of polynomials, Rend. Circ. Mat. Palermo (2) 32 (1983), 157-187.


SOME RESULTS ON FIXED POINTS FOR THREE MAPS

·' By

Ashok Ganguly

Department of Mathematics, G. S. Institute of Technology and Science,

Indoi.-e-452003, M. P., India

tReceived: March 15, 1984)

In this note we present some results on. fixed point theorem for

three maps which generalize the results of Park and Rhoades [ 4] and

Park [ 3 J respectively. First we give the following definitions and

notations.

Throughout the paper w stands for the set of all nonnegative

integers and R+ for the set of all non-negative real numbers.

A point xo i:: Xis called regular for f, g and h, or, simply regular,

if tr:ere exists a sequence { Xn } C X . satisfying, hx2n+1 = fx2n and

hx2n+ 2 = gx2n+1 for each n i:: w, and sup { d (hx1 , hx; ) J i, j s w}

< =· It should be noted that for f (X) U g (X) C h (X), such a

{x,.} always exists.

Let 0 (xo) = { hx,. I ni:: w} , and 3 [ 0 (xo)] denote the diameter

of 0 (xo).

Let 3 (x,y) = diameter { O(x) U () (u) } where x and y are regular

We have the following results :

Theorem 1. Let f, g, h be self-maps of a metric space (X,d) such

that fh = hf, gh = hg, cfo : R+ ----l- R+, cfo non-decreasing, continuous

from the right, and satisfying rp (t) < t for eacht i>O., Suppose there

52 ]

exists a regular point xo e: X such tfiat {hxn} has a clus;er point ao e: X,

whieh,is regu1ar.

If

{I) d(fx, gy) ~ .P(~[O(x) U"0;(1·JJ) for each x, ye: {xn} U{an} U hao,

where {an} is defined by ha2n+I =fa2n and ha2n+2=ga2n+I for n e: w, and

(2) h is continuous at ao,

then hao is a common fixedpoini :forf, .g; hand {hxn} converges to ao.

lj (I) is satisfied for all regular points x, y E X, then ha0 is the unique

commonfixedpoint off, g and h.

The r~ult,can .be easilY proved by couplhg the proof for two ::· •. ·· >•. ·. .' - .- ' . •

maps pyJ~~rk,and Rhoades J4] with the usual technique for the proof . ' " . •) ' - ~ _,: - ,

of fixed point for theree maps, given by Ganguly((IJ,[21) and Singh[5J ..

:;J/heore~ 2. LetJ,g, lzb.e self maps of a complete metric space

(X, d) such that fh=hf _,gh=hg, his continuous, and</> : R+-+ R+,

<j>(t) < t for each t > 0. If every point of Xis a regular point and (i)

is scitisfiedfr:Ir al! x, y "X, then f, g and hfhave a unique common fixed point and {hxn} converges to :the fixed point fm1 each x E,X.

The result follows from Theorem I.

Remark 1. For f =g, \\e have Theorems I and 2 of Park and

Rhoades,[ 4].

Remark 2. From the logic of Park and Rhoades ([4], pp. 116-117)

it,follows.,that Theorei:n l of Canguly [ l] is a special case of Theorems 2

and J, respectively.

No·w .we state a different result which can also be easily .proved.

JJ.i'heorem ,3, Let (X, .d) .be am1etric space, f, g and:h be the same

a;s··ah'ove. S11ppose' there exists a regular :point.xo EX such that

'I\

·,{

[ 53

/;!1) O(xo has a cluster point ao E X which is ngular; ;;,f::~;:

:/ (2) for any E > 0, there exist Eo < E and ao > 0 such that for any

x, y E {xn} U {an} U {hao},

€ ~ a (X, J') < € + ao implies d (fX, gy) ~ EQ, Where {an} is_

the same as in Theorem 1, and

(3) if hao is continuous at ao,

then hac is a common J ixed point off, g and h, and {hxn} converges to

ao. If (2) is satisfied for all regular points x, y E X, then hao is the

unique common fixed point off, g and h.

Remark 3. For j = g, we have Theorem 2 ( C a') of Park [ 3].

ACKNOWLEDGEMENT

The author is extremely grateful to Prof. S. Park, ~eoul University,

Korea, for his valuable help in the preparation of this paper. He also

thanks Dr. S. L. Singh for sending the reprints of his papers menti

oned below. Finally, he feels indebted to Prof. H. M. Srivastava,

Car:ada, for bis kind encouragement and suggestions.

REFERENCES

[ 1] A. Ganguly, On an extension of the fixed point theorem of Kasahara, Math. Sem. Notes KObe Univ. 10 (1982), 307-310,

[2] A. Gangu1y, On an extension of a fixed point theorem of Meir and Keeler for three mappings, Jfianabha 14 (1984),_157-161.

[3] S. Park, On general codtractive type conditions, J Korean Math. Soc. 17 (1980), 131-140.

l4] S. Park, and B. E. Rhoades, Ex.tension of some fixed point theorems of Hegedus and Kasahara, Math. Sem. Notes KObe

Univ. 9 ll981), 113-118.

[5] S L. Singh, On common fixed point theorems in D--Space, Math.

Sem. Notes Kobe Univ. 7 (1979), 91-97.


A, NOTE ON ENTIRE FUNCTIONS OF IRREGULAR

(p, q)-GROWTH

By

G. S. Srivastava and H. S. Kasana

Department of Mathematics, University of Roorkee,

Roorkee-247667, U. P. India

(Received : June 20, 1983 ; Revised : August 23, 1984)

00

1. Let f(z) = ~ a,. z" be a nonconstant entire fonction. Set n=O

M(r) = M (r, /) = max I f(z) I , µ.(r) = µ.(r, f) = maxn ....,. 0 ! ZI =r , p

{ ! an ! rn}, M(r) and µ.(r) are caUed the maximum modulue and ,the

maximum term of f(z), respectively. The concept of (p, q)-order,

lower (p, q)-order, (p, q)-type and lower (p, q)-type of an entjre

functionf(z) having index-pair (p, q),p> q > 1 has recently been

ir.troduced by Juneja et al. ([2], [3)). Thus f(z) is said to be of

( p, q)- order p and lower ( p, q)-order ,\ if

( 1. 1) lim ~:up_ _lug [ P] M(r) r->oo Jn} --[. - -· =

log q]r

p( p, q) = p /..(p, q) = /..

and the function f(:) having ( p, q)-order p(b q and b = 1 if p = q and log [p]x stands for the

56 1

pth iterate of log x.

Definition. An .entirefoncti:on. for which ( p, q)-c:mler 1and lower

(p, q)-orcer are equal is said to. te of regular (p, q)-growth. Functions , :.' '

which are not of regular (p, q)-growth are said to be of irregular

(p, q)-growth.

It is easy to see that the lower 0

(p, q)~type of an entire function of

irregular c.P: ;q):__gro"°th is always zero. Flence the sh.idy of growth

prcperties of entire functions with nonzero lower ( p,q)-type is limited

to funct.iens of, regular, (p, q)-growth. Thus we need to define a new (_ ',"' ~ L {,. ! . f»' ': ~ • ; / ',. ' . •

constant to study the growth of such functions.

Letf(z) be an entire function of lower ( p, q)-order A(b < A< oo ).

A real valued positive function .\(r) defiend on (0, oo) is said to be a '"'. . _,. ;.~· t· .. \ ' . .. . ·, :·-: 1" ' ~ - - ; '. - - • . '

lower proximate order of ari entir~ furiction f(z) with index--pair

'(p, l1)'if

(i) .\(r) ~ ,\ as r _,. oo and

(ii.) A(qf') A (r) ~ :0 ~s r _,. oo

where l\[q] (r) = log[qJ,. loglq-lJ,. ... log r. r.

If suc11 a function e~ists and

(l.J). lim inf . . /'-'!>00

log[p-l]M(r)_ = t)., 0 < I). < oo, .. [1-1].) >.(r) (log I

then t;.. is termed as generalized ,\(p,q)-type of f(z) with respect to the

comparison function .\(r).

The object of this note is to prove the following theorem which

g¢:nerl:lljze_g, a result due t0 Kumar [4].

....j

(

'[ 57

00.

2. Theorrem. Let f(z) = l: a,. zn be an entire function of fower n=O

(p, q) orq<pr >.(b < >. < oo). Then a necessary and sufficient condition

for /(z) t9 be of generalized >.(p, q)-type t).. is that

(2.1) lim inf i/J(/og n) = tl/M, [

[p~2] ]>.-A ,

n-'roo [og[q-I] (-1/n log l ~n J ) . ·.

provi4ed 'f(n) = I a,./an+i l forms a nondecreasing function of n for n ;:::.. no, wJiere

M-

f l/A ((A-1)1-1 I . . " ~ _!___ I fA

L l

if ( P> q) = (2,2)

if (p. q) = (2,1)

for all other index-pairs-( p, q),

A = l if(p, q) = (2,2) and 0 otherwise and '1>(t) is a real valued

funct~op de(\ned by

(2.2) t """ (!Qg[q-l)r )'\(r)-A ~ 11)(9 /og[q-lJr.

3. To prove this theorem we need the following lemma :

Lamma l. [1]. (/og[q-l], )i\(r)-A is a,pionotone increasing function

for sufficiently large values df r, where:{) <; A <! ;\ •

. . Lemm.a 2• J..et.~(r) be a lower proxiniater0rder. of an e~~ire fun<;tion . ' - . ~ ' ' . . . " • . -- ' .. ·' ·.. / ' . . -'! '", ~

with indeX-pi\ir (p, q). Then

Jim 'll(ht) = ·hlj>t-A, O. <. h < oo• (3.1) t ~00 'P(t) .

Proof. In view of (2.2), we have

58]

,d[log (f>(t)] ~ -'- d[log 11- -

d[lo/ q l;-]

d[ (X (r) ~ A) log[· q lr]

Hence,

lim t-+ 00

1 == ·---· - ·- - --·. (A(r) - A) + i\.[ •q ft) X' (r) •

.. ~·

d[log (f>(t)] _ · 1 d(ll?g t] - X A •

It follows that for given e: > 0 and t > to,

( 1 1 '

• ,\-A - 9) <{Clo~. tJ ~ d[log fl>(t)] < (f-A + e: )d[log tJ~

'which gives on int~gtatihg'behveen th~ limits t and ht

(3 '2 (' . 1 ') . <l>(ht) ( 1 )' • • } -- - e: · logh:<log -- < .-- + E' l~g lz. · A-A . . <ll(t) X-A ·

~ :

....• · . :)~_,

.:.,. :·:

Since e: > O is arbitrary, on.passing t~ limit~ in (3.2) we get the requ

ired result.

4. Proof of the Theorem. Let O<tl. <Coo, Then from (1~3), we have

for any•e:;;J:>- .Oand r > ro (e),

; .. (4.1) /~g[p~l]M(r) >(t1 --e) (/og[q-l], /(~) "

Sincef(z) is of finite (p;q}-order, following Valiron · [5], it can easily

be proved that

/og[p-l]M(r) ro log[p-l]f'(r) as r-+ ~.

Thus, from (4.1), we have

~ ['59 .:(. _.'

log µ(r) > exp[P__.:.21{ (t.\~e:) (log[ q ...... IJ,.l(r) }. !

Since 'Y(n) is a nondecreasing function of n, we have for r satisfying

qr (n-l) ~ r < qr (n),

log j On. f + n log r :;> exp[p-2J{(t.\-e:), _(log[ q-;-1,]r).\(r)}

or,

... ·c;r,.c, CP-+1 >··1"-A ( 4.2) X = . .· ..,, iog 1 . n . . >

exp [(>.-A) log[ q- J{-1/n·log_ I an J } J

[$(log[p-2]n) ]>.-A ------;:---;-:-~-~~~

exp[,\-A) log[ q-Ihtog r-1/n exp[P-21{ (t.\,..-e:),(log[ q-IJ,l'(r) }}].

With little calculation and the use of.properties of'Y(t) ancL\(r); we _. -·""l ....

can see that the minimum value of the expression on the right hatid

side of the above inequality is att~inedfor a value Qf r=r(n) given by

(4.3) E[p~2]f (t.\-e:) (loglq-:-ll,.j-'(r)} ..... . ..,-.::0---=---.-----;-7-----'-- - . n . , AL q-Ifr) . - ·- ;1.,··

. ·. . q .. where £[ q fx) = . II expl1lx.

z=O ·"·:':

For the index-pair ( p,q) = (2, 1 ), ( 4,3) red.~ces t~ r· .-- . '-(::. ~:

,.$r) =__!! __ «> r = ct> { . n · ,· ).. . . .\(t1-e:) .\{(A7'"""t) · • ·.'

Therefore,

- [(n) ]A

exp [ t..{log r-i·(r) (11-e:)/n} I

:;:. .

::~· ...

:···

:i

'60'1

_ e [ . (n) Jl - (n/.\(tA-s)) '

or,

{4.4) X > e ,\ (t.\-:s).

For (p, q) = (2,2)1t (4,3) reduces to

(log r),\ (r}-l = n · <=> fog r=., «'I>( ,\(t n ) ). A(tl..;,;.s) · ·· · A.~e: ·

Therefore, we have in this case

ll>(n)

x > [ (A:-1), «

i . . e.,, .): ,:,_,.

. ,\:l (4.5) }{ > (,\-l)l-1 (ll-e).

For (p, q) -::f:: (2,1) and (2,2). (4.3) gives

J-1

log[q-1], )'\(r)= t,1. l s log[p-2] n/.\<=>to~[q-1] r-.tl\~1_/og[p-,2]n/'A).

Thus

X > f(/og[p-2J~) p. . ,. , ' .

exp { ,\ log[q]. ·(re~l / .$! ~

~ ~

I. )

,_,,,

~ [ '1> (log[p-

21 n) ]"

&ii ( _]___ log[p-l]n/,\) . t,\-s

j

or, from Lemma 2,

[§l

(4.6) X > (t,1.-.E)

(4,2), (4.4), (4.5) and (4.6) combine into

(4.7) lim inf X ;> t;./M. n -+ oo

The inequality in (4.7) obviously holds when !,1.=0 and for t.1.=~, the

same arguments with an arbitrary large number in place of (t.1.-e:)

lead to the result lim inf X = oo. n-+00

To obtain (2.1), we show that the strict inequality in (4. 7) can not

hold. Hence, let a number tr. > f.1. be .given such that

. . [ (log[P-2]n) . . l1m mf --- ··-"'----'----n--?oo log[ q-2]{-I/n log I an I}

·:~

J-A t"' • =M

Let us choose a number !Pl such that t.1. <4: til <;: tri&. Then we have for

alll arge 'val ues of n > n0 ,

log I Un F - n exp[ q-2] [ <!>(log[ q-2

]n) J (tiS/M)I/A-A •

Hence, by Cauchy's inequlity I an I rn ~ M (r), we, again have for

r > ro and n > no,

(4. 8). log M(r) >-n exp[ q-2] [ (log[P-2

]n) J . ·. ., . c· I [1/A A + n log r. · l'fJ M) - ..

For(p, q) =·(2, 1), choose r so that n =Mia rMr>. Then, in view of

Lemma 2, we have

.. ·. ·' r . <l>(n) log M(r) >-n log l.Ce A /'fJ)l/A.

. . n + n log cl>( - ) . Al~

)

62 ]

= nlog [(e.\ tr>)l/'A. <P(n/.\tr>)]. . . <J>(n)

( 4.5) oo tr> rA<r>.

In case ( p, q)·= (2.2), chQ()se

·(log r)~(r)-i = Mn * log r;=lf> ( . . Mn ) ta ( .\-l ).\-l ·ta ( "=! y\-l '

.\ .\

then (4 8) gives ·

log M(') > n [log r-(n)('il/ts)lfJ..-1 ]

oo n/.\ log r

( 4.10) ~ ta (log r)J..<•>.

Lastly let us take ( p, q) =:/= (2, 1) and (2,2). Then we choose r such that

ti> (log[ q-1]r/;s)>.(r/es) = log{ p-2] nf.\ -¢> log[ q-1] (r/e&) """

~ cJ> ( log[ p.,.-21n/A) ) t13 '

e > 0.

Using (3.1 ), ( 4.8) reduces to .- ..

log M(r) > n[log r-exp[ q-2] { ~(_!?g[ p-2Jn}.

( 1/.\ } ]

t13)

oo n[. e + exp[q - 21{ '!J~~~1~2~/>. 1-exp[q-2]\ (fl (log[p-ZJn l]

• • , 1 ta I J . l tR I/.\ J or,

(4.11) log[ p~l ]M(r) > (logI q-I Jr) .\(r) {ta + o (1)}.

Thus, (4.9), (4.10) and (4.11) lead to

[ 63

which is a contradiction. Thus, strict inequality in (4.7) can not hold.

This proves the result (2.1) and hence the theorem.

REFERENCES

[I] S. H. Dwivedi and S. K. Singh, The distribution of a-points of

an entire function, Proc. Amer. Math. Soc. 9 (1958),

562-568.

[2] 0. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p,q)-order

and lower (p,q)-order of an entire function, J. Reine Angew.

Math. 282 (1976), 53-67.

[3] O. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p,q)-type

and lower (p,q)-type of an entire function, J. Reine Angew.

Math. 290 (1977), 80-89.

[4] K. Kumar, On entire functions of irregular growth, Ganita 28

(1977),. 75-78.

[5] G. Valiron, Lectures on the General Theory of Integral Functions,

Chelsea Publ. Co., New York, 1949.

Jiianabba, Vol. 15, 1985

APPLICATIONS OF SRIVASTAVA'S HYPERGEOMETRIC

FUNCTIONS OF THREE VARIABLES IN HEAT CONDUCTION

By

R. C. SINGH ClIANDEL

Depart:ment of Mathematics, D. V. Postgraduate College,

Orai-285001. U., P., India

and

B. N. DWIVEDI

Department of Mathematics, Atarra Postgraduate College,

Atarra-210201, U. P., India

( Receive.d :.January 10, 1983 ; Revised : October 3, 1984)

1. INTRODUCTION

Singh [6] evaluated some integrals involving Kampe de· ~eriet's

function and employed one of them to obtain a solution of a problem

in heat conduction: considered by Bhonsle [l]. Some expansion

formul,ae . involving al::ove functions have also been obtained. Appell's functions and ·the functions related to them have many applications in mathematical physics. Here \'<e evaluate an integral

involving Srivastava's hypergeometric function F<3> [x, y, z] of three

variables ([?L P. 428) and discuss it's applications in solving a problem

c·n heat conducdon considered by Bhonsle [1] and. in establishing an

ewansion formula involving Srivastava function .

. 2. AN INFINITE INTEGRAL

Mul~iplying .both sides of the equation [4, p, 74, Eu (4. 16. l)] by

e -z2 H2~(g) ~~4 u_si;g tJ:ie 9rthogonality property of Hermite polyno-

66]

mlals [5], we have.

·J·oo z2p e -,.z2 H2v (z)dz' =

-oo ifn22<~"1'pl I'(2p.+ 1),

I'(p-v+l) ·p """ 0, 1, 2, •••

which, on application to Srivastava's triple series F<3> [uz2• vz2, wJ, further gives

Joo _ 2 [(a):: (b); (b'); (bH): (c); (c'); (c');

(2.1) z2P e z H?.'J (z) p<3> ... · 1 •

-oo te): : (g); (g'); (g'): (h); (h'); (h");

uz2, vz2, wz2 J dz

= _!'(2p + 1) ~re 22('> - p) p<al [(a.·) 2Pt

1 •. p+1::(b);(b');(b"):

r(l +: p _v) . • (e), ~ + p - \I: :(g); (gc'); (g"):

(c); (c'); (c"); · ·J, u, v, w '

(h); (h');' (H"); . 1

where p = 0, 1, 2, ••• and A + B + B" + C ~ E + G + G" +H,, etc. . , • ~ '.1

An appl'ication ofthe aoove integral will be: slfownin out present . . . .

investigation.

· 3~ .APPLICATION.:Tor HEAT CONDUCTION

Bhonsle [l] employed Hermite polynomials in solving the partial

differential equation

(3.I) a<D - K. 02 - K .z2, at oz2

where Ii> (z,i) tends to zero for large value of-'it; and wh'e:n J z I.~ = .. this equa'ti0n is 'related to the probfem ofheat co'nductfoh

(67

()cl> ()2$ (3.2) - = K -

2- - h1 (~ ~ cl>o),

()t oz .

provided that

<l>o = 0 and h1 = Kz2•

The solutiOn ·ofeqtiatipn:(-3~t)·given by Bhonsle [ lJ i€1

00

(3.3) <D(z, i) :ik: I:· !]')r e-:(1+2r)Kt-z2/2 Ht(z). r=O

We shall . consider the problem of determining a function ID (z, t)

iUChtlfat·

(3.4) cp(z, 0). = z2 e -zBr p<Sl ·r(a) : : (b); (b'); (b"); (c); (c'); (cw);

• . (e) : ~ (g); (g'); (gN): (h); (h'); (h*);

. 2 ·2 2 J 1'Z , VZ , WZ ~ ;

Now from (3.3) and (3A), we have

(3.5) z2P e -z £<3> . . uz2, vz2, wz2 • 2 [(a) : : (b); (b'); (b"): (c); (c'); (c"); J

(e) : : (g); (g'); (g0): (h); (h'); (h');

oo -z2/2 = ~ Qr e H,(z),

r=o·

and, by (2'.l) and the orthogonality property of Hermite polynomials

[2, p.289], we find that

. . . f(2p+lt2µ~2,p-l/2 pc3> [(a), 2pil• p+l :: (b);

(3,6)" Qi" .:mi\ r(l + p-1£/2) µ ! . (e), 1 + p-µ[2: : (g); ,," •'•·' ••• '• • T

68 ]

(b'); (b"): (c); (c'); (c"); J u. v, w

(g ); (g•): (h); (h'); (h"); .

Thus the solution of (3.3) of the problem reduces to

00 2r-2p-_l/2 r(2p+l) e-(1+2r)Kt-z2/2 r(l+p-r/2) r!

(3.7) cl>(x, t) = ~ r=O

[

2p+l, pi-1: :(b); (b'); (b"): (c); (c');(c"); . ·] Hr(z) F<3> (a), -2- . u, v, w

(e), 1-p-r/2 : :(g); (g'); (g"): (h); (h'); (h"); " .

where p = 0, 1, 2, .. ., and A+B+B"+C ~ E+G+G"+H, etc.

4. EXPANSION FORMPLA

By (3.5) and (3.6) we establish the iollowing expansion formula

(4.1) z2P e z / F<3> • . . _ 2 2 . [(a) : : (b);· (b'); (b-,,): (c); (c'); (c");

(e) : : (g); (g'); (gR): (h); (h'); (h");

uz2, vz?., wz2]

= ;' ~(2p+ l) 2r-2p-l/2 r=O f(l-+p:::::..r/2) -;:y-Hh).

r 2p+ 1 P+ 1 : : (h); (b'); (b"): (c); (c'); (c"); J £<3> I (a), -2-' u, v, w

L(e), 1-p-r/2 : : (g); (g'); (g"): (h); (h'); (h"); · •

ACKNOWLEDGEMENTS

Our sincere thanks are due to Prof. H. M. Srivastava for.his

suggestions. The first author is also thankful to the University Gran's

( 69

Commission, New Delhi, for financial assistance prcvided to him,

REFERENCES

[ I] B. R. Bhonsle, Heat Conduction and Hermite polynomials,

Proc Nat. Acad. Sci India Sect. A 36 (1966), 359-360.

[2] R. V. Churchill, Operatiional Mathematics.; McGraw-Hill,

New York, 1958.

[3] A. Erdelyi et af,. Tables of Integral Transforms, Vol. U,

McGraw-hill, New York, 1954.

[4] N. N. Lebedev, Special Functions and Their Applications, Prentice

Hall, Englewood Cliffs, N. J, 1965.

[5] E. D Rainville, Special Functions, Macmillan, New York, 1960.

(6] F. Singh, Exp'.lllsion formulae for Kampe de Feriet and radial

wave functions and heat conduction, Defence Sci. J. l1 (1971 ), 265-272,

[7] H. M. Srivastava, Generalized Neumann expansions involving

hypergeometric functions, Proc. Cambridge Philos Soc. 63

(1967), 425-429.

~.:.


INTEGRALS INVOLVING THE H-FUNCTION

By ''".:

Sujata Verma

Department of Mathematics, M. R. Engineering College,

Jaip!Jr-302017; India ,

(Receiv~d :Jul/15, 1984';' Rivtsed J. August 17, 19B4) .. ,

ABSTRACT ..

The aim of this paper is to evaluate two new integrals involving

Fox's H-function. Besides the H-function, Qµr integrands also involve

Bessel functions and Legendre functions. Since· the· H.'.:.function· is.

sufficiently general in nature, a number of other integrals involvin$

various special functions can also be obtained from these integrals

simply on specializing the pai:amefon~. of the.H...,fu~c.t~ons involved in

the main integrals. For the sake of . iUu<>tration;cwe also evaluat~ here··

two such integrals; these latter int~~ral~;~~e/ of'interest in them~lves and are also believed to be new,.

S!

1. INTRODUCTION ;.;!

The following formulae will be required for;evalq<,tting the main . :· :::

integrals of this paper [I, p. 91, Eq. (3.1); 4, p. 3, Eq. :(5)] :· i

(i) r 1 xcr-,\-a (l-x2)~i,u pµ (x) J&(xt) h(xt) dx Jo 'J

,·

2"'-,\-o-l ro+ia)'r(l+icr)'.1"+,,0 '. = f(o+l) f(A+l) r(l+la-!-J-!µ)T(3/2+i-"--ai.-. !_,..~·___,..._!µ.,..-)

...

.:(.

72 1

• 4F5 ; . . , ;-t2 • ·[!+!cr, l+!a, !(,\-H~+l), }(,\+3+2) : .. J

l+ia,:,_!v-f,,1.,3/2+fa+fv-!J.r., 3+l,,\+l,,\t3+1 . .

where Re{µ)< 1, Re (cr)> _;, 1;

(ii) Ioo x''+ 1 (112+x2y-"/2-5/4 exp( ... -:- P.2,oc __ , )1v( p2x 0 , , d'l!+x2 a2+x2

,\- -! 2v -,\-f r(,\+!) --~ = 2- v . P .; r; a . F(,\/2+ 3/4) r(,\/2 t-J t 5/4)

. iF\TXi 1; v+,\/2+5}4;--1!_1, , 2ot

where

R(~) > -0,p >:O, R{,\+!) > 0 R(v+J) > 0.

MAIN INTEGRALS

(i) J: x<h\'""3 ( l-x: )-iµ P~ (x) j ,(x1) J .i.(xt)

. H z xP dx. m, n r I (a;• ·a;)HP l JJ~ q L (h1;, ~;)H!i, J

21"-,\-3-a- l 1,\ + 3 .fr;

- r(3+ t) re"+ D

00

• E ,;;o-r2r ("+~+l ), c+~+2 ), (-12 )' (,\+l)r(3+l)r(,\+B+l)r ·. ·r!'

(1.1)

)dx

'(1.2)

[ 73

H 2-pz m,n+l [ \A] • p+1, q+2 \ B ,

(2.1)

where A = (- cr - 2r, p ), (aJ, a;)i,p,

B = (b;, ~;)i.a• (- ! cr + t v + ! µ - r, !p),

(- ! - !cr - lv + lµ - r, ~F),

and Re (.u) < 1, Re (a) > - 1

Re (cr + 1) t p min Re ( __ b;_) > O 1 <;,j<;,n ~J

(ii) r= xv+ 1 (a2 +x2f>../2-5/4 exp (- __p2 a \_,Jv( p2 x) Jo or.2+x2/ \a2+x2

. H z(ix2 + x'.}" Id\'. m, n [ I (a;. a;)i.P j

p, q _ (h;, Mi,i J

-,\-v- 1 2v -! -,\--1'-2 2 p 1"" a 2

(_ 1!_2_)r m+l, n [ 2 ! E J oo 2or. H (2a) µ z ,

. I: 1 p+2, q+l I F r=O r.

(2.2)

where E =(a;, r.<J)i,p, (;1/2+3/4, µ), (v+A/2+5/4+r, µ),

F = (.\+i+r, 2µ), (b;, ~;)1 q ,

and

Re(«) > 0, p > 0, Re(,\+ 1/2) > 0

Re (v+l) > 0, 2µ max 1 <;,j~n

Re ( arl) < Re (A+i) , or.;

The H-functions occurring in (2.1) and (2.2) stand for the well -

H 2-pz m,n+l [ IA] • p+l, q+2 B ,

where A = (- cr - 2r, p ), (a;, a;)i,p,

B = (b;, ~;)i,q, (- t cr + i v + ! µ - r, !p),

(- ! - icr - !v + lµ - r, ~p),

and Re (,u) < 1, Re (a) > - 1

Re (a+ 1) + p min Re ( ___ b;_) > 0 1 <;,j,;;;;,n ~;

(ii) r= x'i + 1 (a2 +x2f>../2-5/4 exp (- _p2 a \,v( p2 x) Jo or.2+xz / \ a2+x2

. H z(rx2 + x'.r !dx m, n [ l (a;. a;)i.P ]

p, q _ (b;, Mi.1 J

')->..-v-~ 2v }, ->..--~ ~ p -i.· a 2

[ 73

(2.1)

oo - i; H (2a )lµ z , (2.2) ( p2 } r m + 1, n [ ! E J ~ r' p+2, q+I 1 F

r=O ·

where E =(a;, or.;)i.P• (A/2+3/4, µ), (v+>../2+5/4+r, µ),

and

F = (>..+t+r, 2µ,), (b;, ~j)i_q ,

Re(or.) > O,p > O,Re(>..+l/2) > 0

Re (v+l) > 0, 2µ n1ax l<;,j~n

Re ( a;-l) < Re(>.+!) , or.;

The H-functions occurring in (2.1) and (2.2) stand for the well -

74 J

known (Fox's) H-function defined and represented as follows:

H m, n [x I (ai, ai) ... (a'P, a'P) J p, q (bi, ~1) ... (b,, ~q)

1n ' n

- 2~i JL n I'(b1 - ~;s) n r(l - a; + cx1s)

j=} j=} X-' ds, ( 2.3) q p

Il r(I-b1 +~;s) II r(a;-a1s) j=m+ 1 j=n+ 1 ·

wt.ere the nature of contour L in (2.3), the conditions of its convergence

and some of the properties of the fl-function can be found in Chapter

2 of a recent book on the subject [5]. We have restricted vurselves in

the present study to the case where the parameters of the H-function

occurring throughout the present paper satisfy conditions corresponding

appropriately to the following conditions for (2.3)

n (i) A = l:

j=l

p m Uj - l; Uj + l; ~j -

n+l 1

(ii) j arg x I < ! Arc

q ~

m+l ~j > 0

Proof. To derive (2.1 ), we first express the H-function on the left-hand

side of it with the help of (2.3), and obtain

JI xcr-,\-3 (1 ""'.""" x2)-!µ pl1- J 8(xt) h(\t) dx 0 ·. y .

m n Il r(b1-~1~) IT f(I-a1+a1~) . _l_J j=l j=l zl;xplld~.(A)

2rci L q P n re 1-b1 +~1~) n r(a1-cx;~)

j=m+I j=n+I

[ 75

On changing the order of integration in (A) [which is easily permissi

ble under the conditions stated in (2.1)], putting the value of x -inte

gral in (A) from ( 1.1 ), we get

m n

J II r(b; - ~;;) II r(l - a; + Cl.;1;)

J__ j=l j=l zlld; 2rr;i L q p

II f(l - b; + ~;t,) II f(a;-a;;) }=m+I j=n+l

2µ-,.\-i)-l ra + t (cr+p;)] r[l + t (a +p;)] t,.\+I> f(o+ I) f(,\+ 1) f(l + t (c;+p~)-i~~tµ) r'(3/2+! (cr---+ p-t)_+_!-v--t/.i)

n+t (cr+p;), I+! (cr+pt,), HA+ o+ 1), i(,.\+3+2) l • 4F51 ;-t2

Ll+Hcr+p;)~~v-!µ, 3/2+Hcr+p;)+!v-iµ,i>+ l,,.\+ l,,.\+o+ 1 J (B)

N:ow representing the ftmction 4F5 involved in the above expre8sion in

a series form, and using the duplication formula for the gamma funct

ion, \\e easily get after a little simplification:

m n

~ n r(b.;-~;;) II r(l -a;+ix;;)

J=l -- j=l z!; a; 2rr;i L q p

II f(l-b; + ~;;) II f(a;-oc;;) j=m+l j=n+l

2fk-,.\-3-cr-1 /1. + 3 .,r rr;

• r<.o+l) r(t\+l)

~ 4~2r r(l+cr+2r+pt,·)· ("+o+ 1) ("'+I>. +2) r-0 ( --- 2 .r . 2 - o+l)r(t\+1)~ (t\+O-t-l)r . r' ''

1 (-12y r(l +icr-tv_;tµ+r+!P;) f(3/2+!cr+iv,.,..-tµ+r+fp;) rt -

76]

Changing the order of integration and summation in (C) and interpr

eting the result thus obtained with the help of (2.3), we get the

required result (2.1 ). Proof of (2.2) follows on lines similar to that of

(2.1) on using the formula (1.2).

3. SPECIAL CASES

(i) On specializing the parameters of the H-functions involved in

(2.1) suitably, we get by means of a known formula [2, p. 600] the

following interesting integral after a little simplification:

I~ xa-,\-a (1-xzffl-' p~ (x) fo(xt) h(xc)

• 2F1 [a, b: c; zxP] dx

2µ->..-a-cr- J 1A-+ 3 if 1t.

= ----------r(,\+J) r(a+ 1) r(l+tcr-tv-!µ) r(3/2+!cr+iv-~µ)

(>..+a+1) (>..+a+2)

x c; ro+cr+2r) 2 .~ 2 r ----

r=O CA+l)r (a+I)r (3/2+icr+i~-!fL)r (l+!cr-!v-if')r

[a, b, l+cr+2r J -p

~ ;~ c, l+!cr-!v-!µ+r, !+!cr+!v-!µ+r

where Re(!-') < 1, Re (cr) > - 1 and I arg (1-z) I < r;.

(ii) On specializing the parameter of the H-function involved in

(2.2) suitably we get by means of a known formula [3, p, 11 (1. 7. 5)]:

J~ x''+ 1 (~2+x2fA/2-5 /4 exp ( - ~~~x2 ) 1v( S~x2 )

• Kt¥ [2zi (rx2 + x~)µ/2 ]dx

[ 77

2-,\-v+a-3/2 2v ! -,\-! p re a

00 ·(- p2

) r 3, 0 [ 2 I [ l ~ _3a H (2a) µ z r=O r! 2, 3 J J

where I= (A/2+3/4, µ), (v+A/2+5/4+r, µ),

J = {a 2 a 'I ), ( a;a 'I ), (A-;-!+r, 2µ),

and

Re(a) > 0,p 3> 0, R(A+!) > 0, R(,1+1) > 0.

ACKNOWLEDGEMENTS

The author is highly grateful to Dr. K. C. Gupta for his help and

guidance in the preparation of the above paper.

Thanks are also due to Prof. H. M. Srivastava for his encouragement

and keen intereqt.

REFERENCES

[1] B. R. Bhonste, Some infinite integral involving the peroduct o.f Whittaker functions and generalized hypergeometric functioni. Bull. Calcutta Math Soc. 49 (1957), 89-93.

[2] K. C. Gupta and U, C. Jain, The H-functioil. II, Proc. Nat.

Acad. Sci. India Sect. A 36 (1966), 594-609.

[3) A. M. Mathai and R. K. Saxena, The fl-Function with Applications in Statistics and Other Disciplines, Wiley Eastern,

New Delhi, 1978.

[4] P. N. Rathie, A theorem on Hankel transforms, Proc. Nat. Acad. Sci. India Sect. A 37 (1967), 1-4

[5] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982.

,-


ON FIXED POINTS OF WEAKLY COMMUTING MAPPINGS

IN COMPACT METRIC SPACES

By

SALVATORE SESSA

lstituto Matematico Facolta di Architettura Uoiversita di Napoli

Via Monteoliveto 3, 80134 Napoli, Italy

and

BRIAN FISHER

Depa~~ment of Mathematic;~, Tbe. University

Leicester, LEI 7RH, England

(Received : October 30, 1984)

ABSTRACT

In this paper, our main fixed point theorem, given in a compact

metric space; improves a recent result of the second author using a

condition of weak commutativity indebted to the first author. Further,

we prove that c.ther related results, established for commuting mapp

ings, are not extendable to the case of weakly commuting mappings.

1. Main theorem

Let (X, d) be a metric space antj kt S, I be two selfmaps of X. Sessa [6], generalizing a result of Das~ and Naik [I], defined the pair

(S, I) to be weakly commuting if

d (Six, /Sx) ·<;; d(Ix, Sx) (I)

for all x in X.

Clearly two commuting mappings weakly commute but tne con

v~rse is not genernlly tr,µe as is shown in example 1 · below.

80 l

In the fol!owing we study common fixed points of mappings dfie

ne.d on compact metric spaces. In this direction, we recall the follow

ing theorem of fisher [2].

THI: OREM I. Let S, T, I and J be continuous self maps of a com

pact metric space ( r, d) satisfying the inequality

d(Sx,' Ty) < max { d(lx,Jy), d (lx,Ty), d (Sx,Jy),

! d (Ix,Sx), ! d (Jy,Ty) } (2)

for all x,y in X for which the right-hand, side of the inequality (2) is

po~ itive. If

S (Y) C l(X), T(X) C J(X) (3)

ar d S ar,d T CC'mnrnte with l and J respectively, then S, T, I and J

ha:ve.:a tmique common fixed point z. Further, z is the unique common

fi;xed' point of S andland of T and J.

Slight:y.modifying the, proofs of f2J. we can show that

· THEOREM 2. 1 i1eorem I also holds if S and T weakly commuts

.with i and J respectivdy

·PROOF; We give it for the convenience of the reader. first of alh

we observe th2,t (2) implies that

d{Sx, Ty) < max { d(Ix, Jy), d(!x, Ty), d(Sx, Jy),

·~ [ d(fx, Jy) + d(Sx, Jy) ], ! [ d(lx, Jy) + d(Ix, Ty)]}

and therefore

d(Sx, TyJ..~· max {,d(Ix~Jy),,d(Jx, Ty'f, d{Sx,Jy}'} ·. (4f

[ 8\l

for all x, y in X for which the r.igt,~-hand· s~de of inequalit)' (4)js.

positive.

We distinguish two cases.

(i) If the right-hand side of ( 4) is positive for all x; y in X, then the

function

g(x,:y) == ,d(Sx,1Ty) ' . ~~~~~~- -~~~~-

max { d(Jx, Jy), dlfx, Ty), u(~x, Jy)}

is ¢0ptinuous on· the. compact metric space X x Xand so achieves its

maximum value c. In virtue of (4), c <: L This ineans that

d(Sx, Ty) < c. max { d(Ix, Jy), d(lx, Ty) d(Sx, Jy)} (5)

for all x, y in X. Let x0 (resp. yo) be an arbitrary point iiiX arid define

a sequence {x .. } (resp. {y .. } ) inductively by choosing a point x,. (resp.

y .. ) such that

Sx,._1 = Ix., (resp. Ty,._1 = Jy,.)

for n= 1,2, ... This can be done because of (3). It follows similarly. to

the proof of Theorem I of [2] that th~. i>,equences { Sx,.} = {lxn-1-1} and

{Ty,.} = {Jy,.+1} have the same limit z in X. Since S and I are continu

ous, the sequences {/Sx,.} and {S/x,.} ·converge h> lz and Sz respectively.

Further using (1) we have

d{Slxn., /z) ~ d(S!z,., ISx,.) + d(ISx,., lz)

~ d(lx,., Sx,.) + d(!Sxn, lz).

Letting n tend to infinity, we deduce that {Slxn} converges to Iz

so Sz=/z. Using (5), it follows that

d(Sz, Tyn) ~ c. lll.aX, { t/(/z, .Ty,.), q(lz, Ty,.), f[(S~ • .Jy'!) }·. ~· ' ;'.· ., ,• ' . , ' ; . . ' . . ' . ·. -': .. ·~· .. .. " . - . ' .

82.]

and on letting n tends to fofinity we aveh

d(Sz, z) ~ c. max { d(lz, z), d(Iz, z), d(Sz, z) }

c. d(Sr., z). ·

Since c < l, we see that z is a fixed point of S and therefore also

a fixed point of/. Similarly, we can prove that z is a fixed point of _T

and J.

W): Suppose that the right-hand side of (4) is zero for some x, yin X.

Then, as in the proof of Theorem 5 of l2J, we have .

Ix =· Jy = Sx = Ty

Since S weakly commutes with l, we have

d(Slx, JSx) ~ d(Ix, Sx) = 0

and so

!Sx = Six = S2x = STy.

lf s2x ::j=. Ty, then using (4) we have

d(S2x, Ty) < max { d(I.Sx;.Jy), d(!Sx, TJ ), d(S2x, Jy)}

= d(S2x, Ty),

a contradiction. Hence S2x ~ Ty = STy and so Ty = z is a fixed

point of S. We now have

Jz =}Ty ~ !Sx = Six = STy = Sz ·= .:i· ~'.,· . : '; .· ·. . . ' . .. .. '

d

and so z is also a fixed point of /.

Similarly, using the·weak·c~mmutativityof 'tand J, we can pr~ve

( 83'

that T and J have i coilirii<5ri 1fixed pbint ·w>we'now claim thif z = w~ Otherwise ( 4) implie<>

d(z, w) = d(Sz, Tw)

',. < m~x { d(Iz, /w), q{Iz,_Tlk:), d(Sz, fw)}

= d(z; w),.

a contradiction.

Thus in both cases we have shov.-n that S, T, I and J have a

common fixed point z 1t follows easily that z is the unique ·common·

fixed point of S and 1 and of T and J. This _completes the proof of.the

theorem.

The corollary follows easily.

b\'lR.oLLARY. Let ·s a~d J be two ~ontin~~us self~aps ~f a com .

pact'tfietri6"-spaci(x, d) satisfying the inequality

d(Sx, Sy) < d(Ix, ly)

for all distinct x, y in X. lf the range of I contains the range of S and

ifs weakly e-O~m lites '~with 1, theh s amt'! have a: iillique ··cohimon

fixed point.

EXAMPLE l. Let x = [O, 1] with the eu_clidean metric .d and let

S and I be two selfmaps of X defined by

Sx = !x - ~x2, Ix == t x .,.1;·

for a;I x in x. Note that S(x) = [O~ 1/8] C: [O, 1/4] =;:;/(X)and

d(S!x, !Sx) = x/16 - x2/128 - x/16 + x2/32 = 3x2/I28

·: ~'·i2;g-:,, ~(1x:, :~t~) · · ,•

84 1

for all x iJl X. Thu~ S and I weakly commute~ further

d(Sx, Sy) = ! l x - y I (1 - ! (x + y) ]

< l 1 x - Y l == d(lx, ly)

for all distinct x, y in X. All the assumptions of the corollary are

therefore satisfied but Theorem 1 is not applicable since

l 1 1 . 1 Six=- x - -x2-¥:- -·-x ---.x2 =ISx

16 128 16 32

for ally x *·O and so Sand I do not commute.

Results related to Theorem 2 can be found in [4] and [7].

2. Another fixed point theorem.

In this section we present a second generalization of Theorem 1

showing that condition (3) and the continuity of the mappings./ and J

are unnecessary.

We first recall the following result \lf Leader [5].

~EMMA.. For a continuous selfmap S on a compact metric space

(X~ d), the core

00

Y = n Sn{X) n=l

is oompact and SY= Y.

PROOF. See proposition 2 of [SJ.

THEOREM 3. Let S and I be commuting selfmaps and let T and

[ 85

J be commutiag self maps ~ta compact m~tric space (X,d) satisfying

inequality (2) for all .r, yin X for which the right-hand side of inequa

lity (2) is positive. If Sand T a~e continuous, then S, T, I and J have

a unique common fixed point z. Further, z is the unique common fixed

point of S and I and of T and J.

PROOF. By the lemma, the core

.00

Y = . n Sn (X) = SY n=l

is a compact subset of .X: of course, Y is non-empty ~usethe family

{S" (X) : n = 1,2, ... } has the finite intersection property. Now let

x be an arbitrary point in Y. Then.~ is in Sn(X) for n = 1,2, .•. and

so be is in /Sn(X) = S11/(X)for n=l,2, ... since I commutes with

S Thus

00

Jx. ti' n S 11/ (X) Q1· n=l

and so I maps Y into Y.

Similarly, the set

00

W = n Tn(X) = TW n=l ·

=. n ·· S11 (X) = ·y•

;n,=;= l ,1

is~ n()n-:-emptY compact.subset of X and.J maps·Winto W. ; ,.

Since d is a continuous m~pping of the compact set Y x W i~to the

reals>, there exist points z' in Y and w'Jn W such that . ). -. -

d(z' , w') = sup { d(x, y) : x € X, y € W}. = L.

Since SY = Y1 there exists a point z in Y such. that Sz . ...:_ z1' ~ .

86 l

Similarly, there exists a point win W such that_tw = w'.

We then have d(Sz, Tw) · L. Suppose that

. max { d(lz, Jw), d(lz, f,;j, d(Sz, Jw)} > O.

Then we have from (2):

L:::::: d(Sz, Twj

< max { d(Iz, Jw), d(lz, Tw), d(Sz, Jw), _

! d(Iz, Sz), ! d(Jw, Tw) }

<. max { d(lz, Jw), d(iz, Tw), d(Sz; Jw),

.. l L d(Iz .. Jw) + d(Jwi Sz) J, · . · .. :. t fd (lz,Jw) + d (Iz, Tw) l} · .

<. i since sz and Jz are in T and Yw and Tw are in W. The foregoing ineq

uality gives a contradiction and therefore

d(Iz, Jw) = d(lz, Tw) = d(Sz, Jw)'= 0

or equivalently

Jz = Jw = Sz = Tw. (6)

It now follows that d(Sz, Tw) = 0 = L. The .set X = .SY therefore

consist'> of the single point z = Sz and the set W = TW consists of the

single point w = Tw. Since /maps Y into Y, lz = _z ancl since l maps

· w into w, -Jw = w. Further (6)' iinplfos ,z = ~. showing· that z is a

common fixed poi11t of s, T, 'a:Qd J.,The unfquerie~s-Jr i iS e~shy proved. · . . ,_

This concludes the proof. ·-~

.. ,_ .. :

[ 87

To see the condition that Xis compact in Theorem 3 is necessary,

consider the following example:

EXAMPLE 2. Let X = [ l, oo) with the euclidean metric d and

define S, T, I and .l by

Sx = 3x, Tx = 6x, Ix = 4x, Jx = 8x

for all x in X. Since

l. 3. l x - 2y I < 4 I x -2y l = d(lx, Jy) · if x =f=. 2y d(Sx, Ty)= ·

0 < x/2 = (4x- 3x)/2 = d(Ix, Sx)/2 jfx = 2y

for all x, yin X, it is easily seen that all the conditions of Theorem 3

are satisfied except the compactness of X, but S, T~ I and J have no

fixed points;

T}1.e ne.J!,t example shows thatthe condition that S ·and T commute

with J and ]respectively is necessary in Theorem 3.

: EXAMPLE~·. Let X = (0, H U { 1/3} with the euclidean: metric

d and define S, T, I and J by

Sx = 1/4, Tx = x, Ix = 1/3,Jx = x/2


f 1/4 -y < 1/3 -y/2 = d(Sx, J.y)

d(Sx, Ty)= · 1/3 - t < 1/3 - 1/6 = d(f.x, Jy)

if y":f= '1'/3 .

ify,_ IL;J

for all x, y in X, then all the conditions of Theorem 3· are satisfied

except the commutativity of S and /, but S, T, I and J have no

common fixed points

The following example sho.vs that the continuity d'f Sand Tis a

88 ]

necessary condition in Theorem 3.

EXAMPLE 4. Let X = [O,l] with the euclidean metric d and define

Sx = f 1/2 if x = o.

Ix= · f 1 if x = o.

x/4 if x = 0, x/2 if x =j:. 0 .

Tx = 0, Jx = x


f .i < 1 = d(!O, Ty)

d(Sx, Ty) = . :/4 < x/2 = d(lx, Ty)

if x = 0, y € y

if x =j:. 0, y € y

it is easily seen that all the conditions of Theorem 3. are satisfied except

the continuity of S, but S, T, I and J have no common fixed points,

T}le next e~ample shows that theorem 3 is. not extendable to the

case for weakly commuting mappings ..

EXAMPLE 5 . Let X = {x, y, · z, w} be a finite set with m.etrie

d defined by

d(x,x) =d(y,y)= d(z,z) = d(w,w) = O,•

tl (x, z) = d (x, w) = d (y, z) = d (y, w) = 1.

d (x, y) = d (z, w) = 2

and' define S = T, I and J by

Sx = Sy == Sz = y, Sw = z,

Ix = ly = lz = x, Iw = w,

Jw ..:... Jy = Jz =-= Jw = 'j.

[ 89

Obviously Xis c~mpact and S is continuous on J(.

With a routine calculation. it is easily proved that S and I weakly

commute whereas S commutes with J,

Further, since

d(Sx, Sy) = d(Sx, Sz) = d(Sy~ Sz) = d(y, y) = o.

d(Sx, Sw) = d(y, z) = I < 2 = d(x, y) = d(lx, Jw),

d(Sy, Sw) = d(y, z) = 1. < 2 = d(x, ;v) = d(ly,Jw),

d(Sz, Sw) = d(y, z) = 1 < 2 = d(x, y) = d(lz, Jw),

we see that condition (2) is satisfied but S, I and J do not have a com

mon fixed point.

We also note that since

S(X) = {y, z} ¢ {x, w} = l(X),

T(X) = {y, z} ::> {y} = J(X),

this example shows that condition(3) cannot be dropped in Theorem 2.

We conclude this section pointing out that Theorems 2 and 3 are

two distinct generalizations of Theorem l. Of course, in virtue of

example 5, Theorems 2 and 3 are not comparable.

3. A related result.

Using different assumptions on the mappings under discussion

and satisfying a more general condition than (2), Fisher proved the

following result in [3 ].

..

.. ~o. l

THEOREM 4. Let S, T; Pand J be selfmaps.ofa comp~ct :inetric

space (X, d) satisfying the. in,equaljty . .. .... . ''".. . . ~ .

d(Sx, TJ) ~. max { t((Ix, Jj), d(:Jx/Sx)~· d(jy,' Ty),

. d(Ix, Ty), d(Jy, s.\-)} (7)

for all x, yin X for which the right-hand side of the inequality (7)

is positive. If S and T .. commute, if 1 and' J commut~ with ST '" . !.. ' . '. "

and if ST is co~tinuous. then S, T, I and Jhave a. uniqile common

fixed point zi Fii~ther, i isthe•unique common fixed point of S and I

and of T and J.

Note that i~ th_is .. theore.m.the assumption ,(3) · and the continuity ; .

of S, T, I and J are unnecessary w,h~reas.· the ·<:o.mmutativity oft he

··pairs (S,T),·(I. ST) is a.ssumed. . . . . . . - ··,;

We now show in the next exa1Dple that· Thepr~foi · 4 ·fails if only

the weak commutativity of the pair(/, ST) is ass~i:ne4,.even if ~=T.

EXAMPLE6. Let (X,d), S=TandJbe asinexample·s. Define

]by Ix_;_ Iy ='Iz = lw = x.

· it'iseasiiy seen that J commutes with s2 and s2 is continuous.

S..ince

1s2x = 1:S2'/ = 1s~i '= )s2w = ly .= .x,,

S2Jx = S2fy = S2fz = S2fw = S2x = y, .. ' . :· -

52 weakly commutes (but does not com~ute) w.ith I ... . ·· .. · . . .

Further, as alre~dy seen )n.ex,ample 5, condition. (7)holds but'

S, l artd J do not have a common fixed point.

[ 91

REFERENCES

I. K. M. Das and K. V. Naik, Common fixed point theorems for

commuting maps on a metric space, P(oc. Amer. Math. Soc,

77 (19i9), 369-373.

2. B. Fisher, Four mappings with a common fixed point, J. Univ.

Kuwait Sci. 8 (1981), 131-140.

3. B. Fisher, A common fixed point theorem for four mappings on

a compact metric space, Bull. Ins1. Math. Acad. Sinica, 12

(1984), 24)-252.

4. B. Fisher and S. Sessa, On a fixed point theorem of Gregus,

Intern. J. Math. Sci., (to appear).

5. S. Leader, Ur.iform!y contractive foed points in compact metric

spaces, Proc. Amer. Math. Soc., 86 (1982), 153-158.

6 S. Sessa On a \\eak commutativity condition of mappings in

fixed point considerations, Pub!. Inst. Math., 32 (46) (1982).

149-153,

7. S. Sessa and B. Fisher, On common fixed points of weakly

commuting mappings ani set-valued mappings, (Communi cated)

Jiianabha, Vol. 15, 198S

REMARKS ON SQME FIXED POINT THEOREMS AND

THEIR EXTENSIONS

By

D. E.ANDERSON Depart~ent of Mathematical ··sciences, University of Minnesota, Duluth

Duluth, Minnesota, 55812, U. S. A,

M. D. GUAY

Departme~t of Mathematics, University of Southern Maine

Portland, Maine 04103, U.S. A.

and

K. L. SINGH

Department of Mathematics Fayetteville State University .

Fayetteville, North Carolina 28301, U._S. A'.

(Received : December 20, 1984)

. INTRODUCTION

The main aim of the present paper is to show that; some _results of

Ghosh [4J, Khan [6]and Ray [9] are particular cases of ~ell known

results. We also present a theorem, which extends of Ghosh _and Chatt

erjee [5]. Also we remark that the results of Chatterjee [lJ hold for a

more general class of mapping,s.

1. Prelim.iri.aries and Basic Definitions

Definition 1.1. Let X be a metric space. A mapping T_: J(~ Xis

said to be a Banach operator if there exists k, 0 ~ k < 1, such that

d(T2x, Tx) ~ k d(Tx, x) for all x € X. Tis called a contraction if there

exists ;k, O <;; k < I, such. that d(Tx, Ty) ~ kd(x, y) for· au x,"y € X.

94 1

Remark 1. 1 Clearly any contraction is a Banach operator, but

not coriv'etsely. Let X = R. Define T:. R ,.._,. R by 7'{,-r) """ x2. Then Tis

a contraction on any closed interval [a,.b] c (- !. + !), but a Banach

operator on [a, b] C (- 1, 1). A Banach operator may not be contin

uous and may have more than one fixed point. In fact, let X = [O, I].

Define T: X ,.._,. X by T(x) =• 0, 0 ~··tX < ~. T(x) = J, ! <; x ~ 1.

Then T is a discqntinuous ·B.anach ope.Jator. with. fixed points 0 and 1.

Definition 1. 2 Let B be a bounded set in a no'rmed linear space

X and let't(:B) be1its 'diameter. A point x · € B is said to be a diametral

point of B if sup II x-y Ii = a(B). YEB

A convex set S of Xis said to have normal structure if every bounded

convex subset;S1 of'.S which contains more than one point,· has a point

that is not a Oiametral point of S 1.

Lemma 1. 1. [8, Lemma I. 5] A Banach operator T: x,...,.x, where

Xis a complete metric·spa:ee,.hasaJixed,;J:>oint.

· Lemma'l.2.·· [IO, Theorem l. 3] Let c; a nofl..:.empty' weakly

c&nipacf convex' subset of a. normed linear space, possess ·norm'al

structure.

··If T: C· ~ O satiSfies 11-Tx -'- ry 11 >~' ~ [ 11 x - TX 11 + 11 y - Ty \\ ]

for all x, y € C, then it has a fixed point.

· 2. Results.

Ray [9J pr(fved~the followingth"e-0rem.

Theorem [Ray]. Let Kbe a nonempty, bounded closed .and conyex

subset.of a. reflexive Banach space X and let K have norn:iaL structure.

.[ ~~

Let T be a m~pping of Kip.to it<>elf, ,such that .

(A) \\ Tx ~TY'll ~3a_U x-;- y 1J +.\311,[ II~ - Tx 11 + llY,,"""'.'" Ty II}

+ y [ II x - Ty 11 + 11 y - Tx 11 ]

for all x, y E Kand for ·some a, (3, y E R+ (nonnegative reals) with

. 3~.+ 2\3 + 4y1 ~,~. then, T has a .~11iq1:1~}i~ed,p()int.

I:.emma 2.1 lf(3::? 3,.y > Oand,fsati,~fie~·A,the11the:re,~zj~ts a constant k, 0 < k < I_, such th~tll. '[x -'- T2,Lll .~ k II x - Tx. II •

Proof. Settinz y = Tx in (A) we have·

11 Tx - T2x 11 ~ct. ll x - Tx ii +\3111 x - Tx 11 + 11 Tx -T2x 11)

+ y [ 11 x - T2x 11 + 11 Tx - Tx 11 ] . ,.

~ (a + ~ + y) 11 x - Tx II + (\3 + y) 11 Tx - T2x 11 •

Thus [l - (\3 + y) ] 11 Tx - T2x 11 ~ [ct. + \3 + y] 11 x - Tx 11

or 11 Tx - T2x 11 ~ [a + \3 + y] II x - Tx II. . . l - (~ + Y) .

Let . a +:\3 + Y = k. If k >I, then ct + (3 + y ;;;i: 1 -(3 - y,

or rt.+ 2(3 + 2y > 1, and hence 3ot.+ 2\3.+ 4Y >.I+ 2a,+ 2y> 1,

co,ntradiction ...

Remark 2.1 If \3 > 0, y > 0, it follo:ws·. f:i;oni- rLl:~rp;i,~ .. l t)l,ilt

T is a Banach operator and has a fixed point.

In view of Lemma 2.1 it is enough to prove the theorem for the

cases (3 = 0 and y = 0 only.

C,~i;e _1. If,~ = 0 t.hen (A) reduces _to

96]

11 Tx - Ty 11 ~ oc II x - y II + r [ 11 x .;.... Ty II + 11 y -Tx 11 ].

Lemma 2. 2 If y :> O, there exists a constant le, 0 < k <:: 1,

such that II Tx - T2x II ~ k II x - Tx II .

Proof. As in Lemma 2. 1 we have ll Tx-T2x II< a+ Y II x-Tx II. - 1 - "(

Let «1

+ r = k, k < I. Suppose not, k > 1 will imply a + 2y > 1, -r

whfoh ih tum implies 3a + 4y > I + 2ot + 2y > l, a contradiction.

Ify > 0, the resultfollows from Lemma 1. 1. If y = 0, a· = 1/3,

the result is a particular case of Banach's contraction principle.

Case 2. r = 0. The case (3 = y = 0 need not be considered, due

to Banach's contrai>tion principle, and y = a = 0 follows from

Lemma I. 2. Thus we may assume y = 0 and a > 0, ~ > 0.

Lemma 2. 3 If a> 0, (3 > 0, then there exists a k, 0 < k < 1,

such that II Tx - T2x II~ k fl x - Tx II.

Proof 11Tx-T2x11 ~ex. 11 x-Tx ll + (3 [ 11x-Tx11 + 11Tx-T2x11 ].

Thus 11 Tx - 'f2x tl ~ ex. + [3 11 x - Tx n . Let k = ex. + (3_ • Clearly l + ~- I - ~

k < 1. Suppose not, that is k > 1, then ex. + 2~ > I implies

3a + 2(3 > 1 + 2cx. > 1, a contradiction. Thus in this case T is a

Banach operator and has a fixed:point. -

Remark 2. 2. Result of Kirk is not a special case of [9] as rem. arked on·page 904. In fact 3a == J implies a = l/3, the result of Ray

becomes a particular .case of Banach's contraction principle for

(3 == y'.:._ 0.

The following is _proved in [ 4 J.

Theorem B [ 4, Theorem '3]. Let E be a rotund Banach ·'space,

(.(r,;?,7 ,,_ "·.

M be a compact convex subset cf E and Jet T be.a ~elf-mapping 9fM, •. ~,""',/Jf-:[ is c<>ntinuous and satisfies < ' .· · '

1' • ··· L ' ·

;~;;;.,;;,,~~~--·;~·~:~\>::.; ,~: ;,,· l e·f ;, .j ·~ ~ :«, <:;.:,

11 TTA.x-TTA.y II ~ a1 II x-y II + ae.Jl x:;-::-TT-'ttj :h qa 11 Y'7".TT"~ ll\:

, sif',q,,ll ~!T"~~¥JI ta~ II y~fI.\~·lb,~.,::., _, ·l<.·" (C) ··'.,;IX •

where ai (i: !. 1, 2 , ... , S)'are .. no:ri~negaHve t&l nublbe'ts~udh thilt"L~ .. ,: .! •• 5 .. .. .. . . )

:;: l; L,Q, . :i:?r~l!d .T>.X .. ~;,\X; + ('l """'::" /,.;);::(')f, 0 < . .\ ~ 1, ,,; · : .. •• . i=l . , . i • ... .. . . , •.•

.. ~· ; '.,·.:,, $, , : .. . >)·. ., ;" " •"'~, .; ) ,,_ .i,f ; ,, ';.. g $ >!L' :;if·,-:~

for all x, y " E. If TT>.x = T>. Tx Jor 3:ny x ·~ . .M, then T has a fixea point.

it~fuark' 2:3~· Tlie cbtldiH3n';;(c)1and r6t~:ndity of-the :splicl;'E:fa unnecessary in Theorem B. Since Jlis a eonti'iruou~ '·self.I-map of a

I

... ~~mJ3aqt ·;coJ!XrJJ: 1 sY.l>~~t, ,o[;,~r)l,!HH\cp _.spl!:,ce~, F ~1.~.s.~~ Ji.~ffli ,P;9:'1t by

..• ~1chaY~~r.'1tfixe,d ppjpt t.ii~or~Di., Inf~9t.;fC.T)J1q<J .f.'('I;A-.) ;u:c,tlie, ~~~e. ! ~' :· ;.~:,-_,.._·~- '• ... !,·, ;1';,'•S:·.'::\··, r •'-•. '> • ,•. '',;;···~·. ,., •• ~.~ •' •' ";•' .;f,, ,_t·-_,• .:. ·>J;,:~:·

. wh~re . .f(T) ;.-. {x}i· M: llxJ. :- x},.Jnqeed l~t.y c /!Cf,), ~~~:m Xy,::;:: y, , •.... ,.:': ··•'-!;~···.,:·;, ~: ... •· 'i;'" ·~<?1·-~· ,•·. (,,.·. • ~·\i. •.- .. ~. ·-~·'· /../., 'J.ci;, M/.•.!;,.•:·.-.:i)<) ~)-~:··<":..t1.!

... '·.J'fow T;,lj/) ':-. '.~jii:f.O'~~)y'~:Jil'tiitis ·~·-~ F(i'i),'h~ric~f'(t) 1 C:'-·F(Tf). Conversely supppse·'~z~ F"(T);)~ tHeh Ti:z:Oii'/ wiuch•.: in ·t~ni impil~s z = .\ z + {l ..- ,\)T~ whicA ~mplies z >. Tz; Ti,.µs .F(T>.)."C F(T)h,

. ;- ·: ';·.~\!'.~::·\~ ·,._, . . :. \. ·;··-.·...; ·, . ·, ·.' _f~ ·'· :~1'- -~" t / •.

Rein.ark 2.4. Theore~ 1 [I, p. 9J] :can· tie····hferidCd ·th •the

following type of mapping : There exists a constant k, O ~ k ..:::; J, ~-~.t}v .. ··1

such that .. ·· d(TTmx, TTmy) ~"f ;nia~-{ d(x~y), </(x;TTm~),,(/(j•vTI:niy)€_1u:. ·;:.;f

tf(x, TT my); d(y. TT,,,x)} .

for all X, )I E X, Where X and;;; mX,':r::.·y~~e a$ in' (1]'~ , ; • ... . . .. '

In fact the result is true for any contractive type mappin~ [ l ll'fhr .wpich ,the fi~~'1.P9jlltJs ~n,iqu~·: ; {jL. '. ., J\. '\\~.

·· :' '~~~~~~}:R~~' _., !P~, 5~l°t ~fJ,i~~,,;~,i1,iP~, ~11 > Re, 8~,J·~J,~1J&~oirect;

.. 9~tf1

'1efk.~;{~~'·1} with 'the'usJ~i norm;: •,.;

c;c1 .• >, 0 ' _,, , , : • ,,~. 0 i., •.· ~c., \..,"I

Define T: X-+ X by Ta = b and Tb = ti, Xis· compact T is conti'"

nttou.s, 1but T is witliout'fixed pdillt.

Remark 2.6. The statemerit'\'lin:e 1- Tf[r~ ·•p;'~~df':·i~1· int'Orrect.

Indeep, Jet X be a~Y noi:~ed,Jineai.,sp~ce .. Let.B = {x 11 X: ll x II < 1}. '-''~'\.' ,,, .. ~;_. ,,.t, ~,,, .•.. ;.·;~<:. --:; ,·_;, ·.c ~ --"-~ ,,,_.,;,.-.. '• '· ' " ·' ~. '"

Clearly B is convex. Define T: B -+ B by T(x) = x/2 + a/2, where

a ::j::. o and , 11 · a>!I .:_· .. I; .. ·Clearly t.' is nonexl\an~iVe111and satisfie's the

conditions of T1 [1, p. 90]. However, T.does not have a fixed ·poi~t.. . ·"· . . - . '

Definition 2.1. , A metric space Xis said to be metrically ciikvex

~H or .. ~py x, ;Y " ..Y, with·.~ .. =IF y there fXists ~ " X, x ::j::. z =A .JI , s.l;!clJ, that

1'~·~·;,i:t. d(~,. y) '. 'ti<f~!).. . . . t • '.•:. ''".,.. ·:,·· •. ··-_" • i \;

· .· .. : b~'fiuiti_;~ ';z;?~ :~et KJ>e a:·ndn~pt§'cldsed' sd;bseFdf a 'Irietrrc ··· .~i{~~ x:itti~f'.1~t\s~'_T_l16 m~ppings'dt:K ihfo cn ;cnie'set bf no,tietrtl>'fsi

cl~~edbhhiiJl&~ s'oiisets ofi).' :f'ff~n es.T) is''said' fo'be a '•gt'f!ef'alif&U , .. ·' ,,..._

c.otztraetion .pair.of K into <:"B,(X).~f .there exist nonnegative reals at,~. y ·;1;4~·+: 2~ +·2r.d' 1 ~ti~h th~;·f~;,.anyx;y ">'.K .. ·· .. · ·: · ·.·_ .. , ··-·-· ;< '.. ' - . -- ,. ~. f.·, •• -· • • - ' •• • . • • , - • ) ~(.·."·-~ "··; ,l" ·- l

iicsx; fy).<:.'·i:id(f,yf":+ (f[D(x, si'f:f''n()i,:·r.Y>r+ •,J }

.11[D,(xd)'.Lf D(y, S~)l· . , , : ···" '..." > ...:~ .. -::o._.~~

·NStl' '~ ,;;,1: ·'" • ·:'"· J:~ .. ·· ... : .. · ...... _, .. :~-1 . . , •., ':" ..... ' -~: h.···· . ·,,_ ",:?£:

,:\•

. For any nO:nempty ;subsets A:~, .Q of X we: define ) , , . .

D (A,B) = inf d{a,, b) :. a.,~ A !lnd,b ~- B} ' ,.: " . -~-: " .':, '\.•; . -- .. '· , .. --.~ " .. ..... ;;·,_,,;J ~ ..... ..... :-.;:

and '.: .,j(. . ...

' . ,. · H(A, B) = max { sup {D(a: B) ·~ dflE'.-A.}9 ·supf8(A/•bf !~4; ti 1B}'~.

,:'.~'F'"c -~~ffGeir~~ o! [6,' tb'.e~r~trl.' ~. ' t' f L~t i b~ a c61li1'1~t8'1lffi'{f'i;metri-

'.~~f9

cally C.~l)ve~,,metri~. ~P~F~~;J\;~. :PqP.Cf~lP!Y c_to~d;~~pse~ pf¥,,.L.~l f.§;r:O be a generalized contraction pair of K into CB(X). '

, 1:.;;,;j

any - 1Y .,.. · ,_ -· t ,: ·~--. · ·. , .. t;-~ .. j -- · ·

x Ea K, S(x) c K, T(x) c K and (a+~+Y) (l+~+y)/(I-~-y)2 < 1, the.a there exists z e K snch(th~t z,e 5;(z}iand Z) e T(z). h

Theorem E (12, l'heprem 3. 2] •. Let X be a cPm,plete. metrically :_:; · · .. -:· \1-'"' ·:·:".\;-~ ,- \: .. ,. ··: _(;;-;:. -~---' tz.:~;;i·1_ 1 ·~:·-:/ -:):'.·! -~·::H·."f

convex metric space, K be a closed subset of X, and S, T be two mappi-

ngs of K into CB(X) satisfying the 1bouridary·con.ditiotf;:·Sx c ?K,

Tx c K, (V x e o K). Suppose there . ex.ist no1;megatjve numbers :· '.'-.f. ·! · ~:- :_·,::.._·.;I , . .! .... ,: "1 f~ ;~ .?

a1 , •• :,a~, (iat < l) with

ai + ~~- < ; - 05 , (i =·-1~ 2; j·:··:3:··4)such t~at ·~ ..... :.+. a,5 ··> •.'.• ·''· . .,., ,.,_, ,}: '.,. :t

f;;(S;x, Ty.) ~ 91 d(.r,;~x) j:'!z_~d(Y.r1'ftt/~afl(f.•l.>::l f :1~~r4~· Sx)

+ a5 d(x, y) for all x, y e K. r

. Then t.he fixecl point set ()f each ~. T is n<'nempty and these_ two sets '~~inc'ia~. '·''""· .. ,,, ... _.<::·::·· ·c.·'·.<··· · -'·, ... ,· .. ·A >·' ·;'.'~~nv:.i

: '• ·~~ .1 :;; . ;_ ":· ';., , .~ ·> ~ij'~' -._" •t~j: ;\. ;.,. :\-.

; ,; ; I,.;.,mm~:l~4. ,J]1¥~:C<>P,.4it.i.enru~,~:; "t. !):+.r)(l :+-.. ~ + Y) . 1 . ·. .·. , . ·~.~ ·''?(.fl·::;;.;.;._ R _..;;. -~2rr~~~£f;f:+1:i'!i!~~f

.·-.. --implies

L-:: ... ·,,.:) ~ ··.:: ~:

a; .. ~.a; < ;,-:5 ,(i ~ 1, 2; }'_3, 4) . ...... ·' .,;~ ·:·· '"· . ± ..... 5_.·, ·..: --~-:~ ·:-~ ;,., .. :,.., . .:· ·1- ..i, ....... ,.. .... g:;~: .'<';t

":' .<;; """"'' \ " "1

r (« + :~ :f' y}( 1 + !) + Y): ''.<' '1 impltes < '·' i .;_; G.E

Proof. (I • A~ "''2 . • • ··.•,. ' ... • ··· ·~ ~- ~ t" , .. r v -~ . ~ · ...

···" -~.

"+~+r+«~+~2-1:r~+·it-Y+~r+r2 1<' 1.:;..;:.215..;;..2f +''!)~11f1·1~+·~~r ~-,,

'JOO]

Thus? 1:i1+«~+<t{f3~~t31y <: 1!'if:et<lf!2ds,'!1=af+a2:,'y=:ia3+a4., ;· _;

Then we have,

as+as [a1+a2]+tJi> [as+a4] + 3a1+3a2+3a~+3a4 < 1~ .. . '

r. ";

+ 1 .....,,,. as c·· · ·1 ·"" · ·., .. 3.·. '4>· • a.. a1 < · ·. 1 =··.,.'•; J··= , · / . 3 +as .

':~ri be written as j(at+ a1) + ~s(m + :a,r <'f...:...; 'as y

or a;; + 3(a,i + a1) + a5(ai + a1.). ·< :l .;, 0

,{, ,,)"."~'>,~:;',''; j : : ·' :~,· ·i,_; c-,,:_ , j:··' :'', , -, -:, .

(i) ,,If' l = 1, j = 3, then o5 + 3(a1 +' oa) + a5(a1 + aa.) <: L ,.,,·

(ii) If I = 1, j = 4, then as +3(a1 + 04) + 05(a1 + 04) <, L.

(iii) ·u i = 2; j = 3;then': 05 -t~ 3(02 +" aa) + o~(a~ + o3) <:. 1.

... ,~ ·- ,"'- . . ',_,,' . ·;,'.· .· ;'< .- '.- -_:'""' .. ' ,· ' ; -.. . · .. ··: .. ~~'.: :·-·_::- ·.,,- ·'· .. .;::~:. ·. ·-., , . ..-·;

(iV}7 If r ..:....' 2,·]~:4, then a!i+ J(a2 t ·a4) + aa€a~·+ a4)'<:,r..

Tlt us the Lemma.

; 'Usi!~g symmetric properties of a m~tric, It c~h beia'sii~ se~J ihat

Definition 2~~ is .e<!~ival~nt to the condition of Theor~m E. . ~,, > , .·'.!

'•, ( . . '. ~ ." :: ~ : ·. ::' ') ' ' . . . "' . ,,., . ,-. .

'Jle~rk,':~·7L ltJQlle,ws from 1 Lemm~~.•Mhat iThM~m 3Jl (6J is' '- .. , ... , ..

a special case of Theorem 3.2 (12}. .•'.,

The following result extends Theorem 5, 6 and 7 of [5].

Th~rena t. Lets be a compact metri~spa;:~n°d T bea. contin

uous mappi11g of s_ into· itself.,. $uppose tb~re. e~sts ,a ~ ~mily of functions

F = {/ai}CI E s' fro iii (0, I 1 into s ·such-. tbatfo,i- each a E S, ?ai\l) •' a

arid for.a 1functiop ~F (O~ l)·~ .. (O, Oi d(f r,x (f), fr,. (1)) ~. , .. ,

Ll01

¢ (t) max { d(x, y), d(x, fr., ( t)), d(y, fry lt)), d(x frv {t)), d(y, frx (t))}

for all x, y e S anp for all t. e [0,1 ]. Further, assume that for t --+ to

in [O,lJ and a -+ a 0 in S, f., (t) .....,. f (to) in S, then T has a ao

fixed point.

Pr9of. For,each n;=l,2,J, ... , let kn= 12

;1

, and let

Tn: S.....,. S be defined by T .. (x) = fT.,(kn) for all x e S.

Since T(S) C Sand 0 < kn < I, each Tn is well defined and maps S

into S. Now for each x, y e S we have d(Tnx, Tny)=d(frx(kn), fr 11(k,.))<,.

cp(kn) max {d(x, y), d(x, fr,(kn)), d(y, fr11(k,.)), d(x, frv(kn)), d(y, frx(kn))}

¢ (kn) max {d(x, y), d(x, Tn(x)), d(y, T,.(y)), d(x, Tny)); d(y, T,.(x))}.

Thus each Tn satisfies Ciric s condition. Smee Sis compact (hence

complete), it follows from Theorem 1 [2] each T,. has a unique fixed

point x,. E S. Since Sis compact, there is a subsequence {x n5

} of {x .. }

such that x --+ x e S. Now T x = x --+ x e S. n1 n; n; n;

llsing the continuity of T, we conclude that Tx -+ Tx. . n;

Finally T x. n; n; =fr (k .)-+fr.,(1) = Tx.Itfollowsthat Tx=x, . xn; n,

REFERENCES

[I] D. Chatterjee, Generalized contraction principle, Int. J. Math.

and Math. Sci. 6 (1983), 89-94.

[2] L. B. Ciric, A generalization of Banach's contraction principle,

Proc. Amer. Math. Soc. 45 (1974), 267-273.

[3] W. G. Doston Jr., On fixed points of non-expansive mappings. in

nonconvex sets, Proc, Amer. Math. Soc. 38 (1973),155-156.

1021

[4] K. M .. Ghosh, A generalization of contraction principle, lnt. J.

Math. and Math; Sci 4. (1981), 201...:.206. '

[5] K. Gh~sh ands.' K. Chatterjee, Some fixed point theorems, Bull

Calcutta Math. Soc. 71 (1978) 13-22.

[6] M. S. Khan,, Common fixed point theorems· for multiva!lled mapp

ings, Pacific J. Math. 95 (1931), 327-347,-

[7] W. A .. Kirk, ,A fixed point t~e~re.rµs fer mappings which do uot

i:r;crease distances, Arna,. Math. Monthly., 72 (1965) .1004-

10C6; u

fo1 S. A. Naimpally, K. L. Singh and J. H. M. Whitfield, Fixed

points and nonexpansive 'retracts in locally convex

spaces, Fundapienta Mathematir;ae (To appear) ..

[9] B. K Ray. A fixed' point t·herom i11 Banach spaces, I1~dian J. Pure

and Appl. Math. 8 (1977), 90J-907.

[l O] S. Reich, Remarks on fixed points, Atti Accad. Naz. Lincei, Rend.

CL. Sci. Fis. Mat; Natcer. (8) 52 (1972), 689-697.

[ll] B; E. Rhoades>.·A comparison of variousidefinitions of contractive

mappings, Trans. Amer. Mmh. Soc. 226 (1977), 257·-290.

[12] Do Hong Tan and N. A. Minh, Some fixed point theorems for

mappings of contractive·. type,,· Aet~' Math. Vietnamica 3

(1978), 24 42.

Jfiiiniibha, Vol 15, 1935

ON j V, >. jk 13UMMABI1.ITY OF ULTRASPHERICAL SERIES

BJ!

w, T. SULAIMAN

Department of Mathematics, College of Education,

University of Mosul, Mosul, Iraq

(

Department of Mathematics, Heriot-Watt University,

. Edinburgh EH144AS, U. K.

(Received :March 29, 1982; Revised [final]: November 15, 1984)

In the present paper, a new theorem on I V, >. Jk factor$;; of

ultraspherical series has been proved. The work is motivated by the

recent papers [1], [2], [5] and [6]~

. ' : ' .. -·- ,,.,' .· /''» I. , .Let. :2lun be1a given series with the seq'uerice;ofpa'ttial sums' {S,,}and

let .\={An} tea monotonic non decreasing sequence of-na'.turalnumbers

with .\,,+i-An~ 1 and .\1=1. The sequence to .sequence transformation

·n -V,,(.\) =J/.\n · ~ .· · Sv

V=n-An+l

defines the generalized de la Vallee Poussin means of the sequence {Sn}

generated by .>.. ·

The series ~un is said to be summable I V, >. I , if ,the sequence { ••. ' i

{V .. (.>.)} is of bounded variation, i.e.,

00

~ I Vn+l (.>.) - V,,(>.)' I < oo_

n=l

We say that the serie~ ~Un is summable r v, >. [,, ' k > 1, if

~.\nld I Vn+i(.>.) - v .. (>.) JI' < oo.

104J

On taking ,\,, = n, this summability reduces to I C, 1 j,, summability

and for k = I this is the same as summability I V, ,\ I .

let /(0, if;) be a function defined for the range 0 ~ 0 < ~and 0 ~if;< 2~. The ultraspherical series corresponding tof(0, rfo) on the

sphere Sis

/(0, if;) - 1/2~ ~ (n+,\) J J f(B',if;: ~,.<~>~cos w) sin 0' d0' d<fo' n=O S [srn~ IJ sm2 (ef;-Cfo')'t-2~.112

where

00

~ Un

n=O

cos w = cos 0 cos 0' + sin 0 sin 6' cos (¢-if;').

A generalized mean value of /(0,<fo) on the sphere Shas been defined

by Kogbetliantz [3) as follows:

f(w) = 2~ (si~ w)2.1. J Cw f(0', if;) ds'

[sin20' sin2 (¢-¢ ')J<l-2J..>/2

where the integral is taken around the small circle Cw whose centre is

(0, if>) on the sphere Sand whose curvilinear radius is w.

We write

<fo(w) = /(w)(sin w)U-1

1 Ix q,,.( r) = -- (x - rr-v ,P(t) dt, r(p) o

p>O

I Ix .Pk, ,,(x) = - (x - t)k<v-v t/;(1)'' dt p > 1 - (1/k) I'(kp-k+ I) O '

[105

<l>o(x) = ,P(x), ¢1,(x) = r(p + l)x-11 4>1{•).p > 0

<l>p(.\) = d/dx <l>p.,.1(x)

logkx = log logk-Ix , ... , log2x =log log x.

In this paper we prove the following results:

Theorem 1. If ¢(w) is of bounded variation in <~i. it), whert

s:.;• ,,., 1 - .\ . 'll =. , ~ =A. O <>.<;;ti

A ,\" n

p. • is a large constant, and ·;f

I: I rf;{uJj 11 du = O{tl+C,.~1i11(tog,. l/t)ll}, ast ~o. ,~ (1.2)

« = 2>. + l - A.-~ # o, k and I" are integers such that . A

1 <; k, µ. <; oo, then the series

E /An Un (t)

[log,. (n +. 1)]~+8-! att~x is summable l V ,>. I t, provided {#'a} is a convex seguence iuch that

E n l'n (log'" n)l/Z-1 ~n2 <: oo (181<;1)

and

n (log,. n)l/2-• /j.Ji.n · < oo. E An

Theorem l. If cp(w) is of bounded variation in (lJ,i;) ~here YJ=l"/n•,

b. is a positive real number less than unity satisfying

106]

~ > !::, > l + 2;\ , (0 :::;;; a1

:::;;; p < 1) A ·3+p-0t ·

µ' is a large constant, and if

cf>k, "'(t) = 0 {tH-1cl3 (log!' 1/tp), as t 7 0, y ~ 0, k integer, . ~:::;;; k <' ~ . ''J • ··ci'• ' ·. • .• ·, (1.5)

~hen (he series.

µ,n Un (t) :E(!ogr• (~+ l)ll'+&-1/2

at t =xis summable IV, A lk:provided {p.n}\ii 'd diinVe1i'.s~r.ja~hife iuoh

that (1. 3) and (L 4) hold. '7i

"~·'

2. We require the following lemmasi •·f

Le:mm~,J~;. ,cl{ff con4i[/(Hl~. (1. 2)pn_d(1,(5).imply,re,.spectively,· that

t . . ·: LI <f;(u) I du= 0 {t"'(logl'- l/t)l>f1'} '.''•·

. •·.· (2; J) , . "• ~?. r

and

'cf>i. "' (t) = cP"' (t) ==:= 0 {tl+f3 (lOgl'- lf t)Yfk}, (2.2)

Proof. Fork= 1 the lemffia.foUows directly from{t.i) and (r. 5). ·

Using Holder's ~neqµality we have, for l ·.::::;: k <·oa, .. '• .J'' ' ; -, ,, ' ;

J

t . {JI r!k '}rt {I-<llkl d cfo (u) I du·~ JJ"' (u.) ,,kefu) ·· lJoduf'; · ·· . '\ ..

= 0 {t"' (logl'- l/t)f3/1:}. '\;

By a similar proof' we obtain (2.2) ...

l.emlml, 2.. If {p.n} is a convex ;~qu~~ce such' th~t

then

n µn log n < oo, ~ ,\.,2

m u ~ logP- (Ii + l). 6µ .. = 0 (1), m-+ oo

n = 1 j

f,

nt • . . ; ~ n log;. (n + 1). f22µn = 0 (I), .: m-+ oo.

n = 1 ·

[1;~7,

·. (2:3)

(2.4)

P~oof. ii1~"8~iivergence offz .~.µii ~qg- ~ in;i.pli~l1ne ~on:Vergerice . . . ltn2 ' .. • . , . . .. ,., . • - , •• .. .. . •· -

:,;

of ~ µ .. log n , and the Ia~ter implies µ .. log n-io and n logn An ~.-o.' ·. -~ n

Now using A!:>.~l's transf'{>rma'l:iqn; we 1,iave · · i' E · . 1 ,:: • · · '1.

m

~ "'" --= n=1 1• .• n

m-1 [ n ] [ m ] n,:1 .,.:1/{!" (~µ~).+ .· r;I l(r Um _. ' ' '. i "'

which implies ·. (

., -·

m }:; log n f2pm = 0 (1).

n=l - · \

- /

Again, by Abel's transformation, we have

~ log n 6µn ~' ni ~ }.~T' '~-- loJ r ;]·_ L,~ ;µ. n + n-1 n-tL r-:1. . .· ... , - .•- [-'~ I~~ rl6f~

r=l ·- .. J _. ·

which implies

m }:;

n=l n log n L,2 p.n = 0 (I).

·\, {

168J

3. Proof of Theorem 1. Let Sn te the 12th pai tial sum of the series

(l.l). Then we have (Szego {4], p. 84)

s .. = r(A) In. n _r..,..(~-)-r(-=-1 + f)- f(~t) ~ (k+A)P"ui (cos w) (sin w)2.I. dw

2 O k=O .

= - r(>i) - JfCfi'w>[d {p1A1 (--)+P111 < }l< r(!) ro + ,\) I, . dx n+l A • .. x) ! sin w)2Adw 0 . . . . .. J

. = - r(A) I" !fo(w) d f p<AJ ( } · l\H r(i+.\)° O dw · n+t cos w)+Pn111 (cos fi) . dw

. = s!'1'+ $,.2, say.

s 1 = ~ r(ll { JYJ ["} .. ruff<~+.\) .. 0 + J'lj = J1 + J2. say.

Ji = 0 (n2Hl) I: tf> (w) dw = 0 {n2A+l-& ta-+ ll (log,. n)"'"}

= O { (Jog,.: n)ll/k}.

J. = __ f.{A) . Jrc .. ·. d . 2 r(!) r(i+.\) 'l tP (w) -d~ {P~!'1 (cos~')} dw

f Jrc/2 J"-'lt l" } d = A + + . tf> (w).-d {P~!~ (cos w) } tlw '1l n/2 n-:-YJ w

= J2. 1 + J2. a + J2. a. say·

[ '"'.rc/2 Jrc/2

12, 1 = A rfo (w) P!!11 (cos w) J'1l ..... A

11 P~!i (cos w) dtf> (w)

= 0 (nA-l+.41) = 0 (1).

ii'·;

:;- :· ':". . ~·: ' ... ·-:~'">-~ .[10)

J,, 2 = A[$ (w) P:~\ ( oos 'w )]:~:- A; r:; P:k {cos ~) a;; ( w)

, .... .... ::-·

A [;; (•-w) (-!)"+I~::\ (c~s ~)]:/~; ~ ··. "

(-: 1 )"__A J://;)~~-i. ~~:?~)f);~,~,(f.··~dfh p~~).-;,.~)~.~~':-:: .?:.(~).·; . . :, \ ·_:-.~·~:·;; /::: ,._:;.:.::···.,..:·:1:

. . . r ' . .· .... · .. ·l<t. . . -r:lt. . . . . . J2, a =A. l¢(w)P~i>Y '(cos 1-v) r :_:_A P~~L (cos w) d¢ (w)

. J'lt-·IJ 7"-":i'J : ; ,:-' . : •· . '.: • :: . ·: ::-- ••

= 0 (n2).-1) = 0 (!). I ... ,,·

Therefore .;:;

J2 = 0 (1) .·.·-· c·_-;

and -~ ·,

s,.1 = 0 { (logl'- n)B'k}.

; ·'other prirts ~kthe ~u~ ai:e treated ~ih;ilarl~, . and con'.seq~~n~I; we have .r•:

\ .;

Sn = 0 { (log1• /~)a/k} ·i-..:;

a1id. ;~ ·, , ~· .:~· ..

n ·-· .~ J Sv (~) \k F 0 {1f (ll4gif n)~k:• "" v=O . . ..

:·::-._;;: ·>.· · .. <.

Now let i" ·. ~ ":· \'.

Tn(x) 1 ,l.l

- n+l"v:l vu~(x). "

11.0)

By Aµel's transformation, we have

Tr.(x) = Sn(x) - _J_ ~ Sv(x) = 0 { (log,. n)llfk}. .. ·.•. n+l v=O

which implies ··.·

n ~ I T o(x) f1'1 == 0 { n(Iog,.· n)ll}; • ·

11=1

Let Cn==Vn+i (.\; x)'.;_ V' .. (i\; x), where Vn (A; x)'"is the nM de la Vall~

Poussin mean of the series

I: Pn Un (x) [log,. (n+ l)]P;.1-112 •

By an easy computation, we have

Cn == n+l

_I. .I: {(.\,Ht - .\,.)(v~n-1) +.\,.} An An+t v=n-.\,.+2 · ·

µ.,, u~ (x) [log,. ( v+ 1 )]ll+s-j •

Therefore, in order to.prove the theotem, it is sufficient to show that

00

I: .\!""'1 J Cn ji <: oo. · n=l

Let (i) . . (ii) I: be the summation over all n satisfying An+t = An and :E n n

be the summation over all n, where·Xti+I >·>.,. •. · , .. .

When .\n+i = .\n .. Abel's transformation ~ives t~at '."'.··

111)

1 [ ,, l ., .:·1· .{· . . . --: . ·1· ,· ·. Cn == -- l: E rur (x) A' l'v -

An+i v=n-A,.+2 r=l . v[log,.(v+l)]llH-1/2

"'n-i\n+2 -·- ___ · _. {n-J\~+.l ru,(x} } -· -· • 2 r=l. , · .·

+, · pn+i ln+L }] · " •• ·en+ 2)Jll+l~l/2 · ~; ru,(x}_ -. r=J .

,7"' . . L!+ L:+ L!. say •.

By Minkowski's inequality it is therefore sufficient to prove that

Now

(0 . . . E ""-1 l L~: 11: < oo for r=l, 2, 3. n .. n

(i) ~;\Lt

n " IL! I• = 0(1} ~} _!_

n i\n

D. h{log,.( 11:i~)Jl+1-112 } J [

n . E_ v t Tv(x} f

v=n..;.:..An+2

. (i) , n . l /As l = o (1)~ _! Z v}T~(x) /,. .6 v[log,.(v+lJl a,.1 i12 J

n ;\,. v=n-.\,.+2

. - 0(1) •~I 'I T.(x) I' e.{ •[lo~v+':)1o+•-H•}: •:!:-l A~ = 0(1) ~ 1v I Ta(x)I,,. A tv[log"(v{;na+r~l/2 J·.· ..

Using Abel's ttansformation agal~, by (i.i), ~~easity ha~e

i:

112] ~:.;

~··

·.,

(~;:-; ·. I I!.' I ;-:- 0(1) • n ~7'(1og•n )~ /' •l i<[log>(n:;) I OH 112} n ·'· _,_. - -.... 1 ...

·~~. ~·, ~,· ~~ ·.'.>';··

'o (1) ~--· ,n (log"n)(l-2&)/2

t:,2µ,n + 0 (I) ~ (log:•n)112-& h,µn {li= 1 :· ·· ·-··: , ... -._. n= 1 ' ~.. c- . _,, . -;:,~ :-,·_ ......

, 0,0 1 . + 0 (1;) - :S --;....:·· µn (log"" n)l/2-B · · n=l n

., ~·11 ••

.. -;-: O(J),, .. · ., ...... · ··-·:. ·d:2)

by (2.3)'ana (2'4); and hypothe~is (U):

Further, applying Abel's transformation and (3. I), it is easy. to

see that

~-

.-.!\

(i) . - 'di) ·. . . ·.· '.' : ... »::,x1i:;,1 c /' L 2 /k +.,.~ ,\Tof-1,.,, L,.3 /k n n ' 16 •.... : """;'.''j,i .. ?1'· "

;.: li.

:: ~ :· ., \' ::: ...

:\: :.~: .'· !:

00 ·:~ ·i .... '.h ·:· .·, 'µ~~- ... ::· 2 -= 0 (I) :s

1 r T,,(A') jk - 4,.[Jqgf(~·+12w+&-112 n= ,,

.. ·!t>· . :··· .~· ··oo· . ·:·· =i ,Q (I) .:S 1!._~J:?l:~lng""n )112"'&

, .. , - ' ' n~l An '·i:

:··.:~ ~:::· ·.

;~ ~ \!I.

. -···· 00

+ O'(l) 2: · .. n~J·_

ffp.n (log", n)I12--s:'' ';\~2 ---

00: . + q:_{l) 2: · µ7 •. (log"" 11)112-s n·s~( .· ·. .\n , -

•.: ~ .

;:·; b (1): ·~- .

'···,..-·.

: ~~ ir··

·.:.:_i./·'..·

(3:3)

[113

by hypothesis (1 3) and ( 1.4).

(ii) . . . Now, in order to estimate :E we have, with the aid of Abel's

\ ·n '\!~(\ trapsformation,

·~;~;~~i: 1 r n . .

/ Cn I ,;;;;; --L :E v I Tv (x) l . An ..\,.+l V=n~.>.,,+2

\' D.{(t\ 11 + v- n - 1) v[logl'(v~\) ]PH~112-}f · ·· . µn-.>.,,+2 ·

+ (n-.>.,,+ 1) IT n-.>. .. + l(x)j ln-ll .. +2)[1ogl'(n-ll~+3)] P+s-112

·l + (n+ l) 1 Tnq (x)j (n+ l)[lo~:(~~\)fii+s-112 J = M~ + M; + M!, say.

By Minkowski's inequality, it is therefore su1icient to provethat

(ii) :E >,~-1 I M~ F' < oo for r = 1, 2, 3. n

Now

(ii) . . (ii) [ n :E .>.L1 I M 1

·11< ~ :E - 1 :E v I T (x) I " n ""=: k+l . v . n n ..\ V=n-.>.n+ 1 · ..

n

l..\vl:,. C[logl'(v:~)]lho-1/2) + v[!og"' (v~vl)]PH-1/2 }J ([

(ii) l n . ·.. · ~ 2: _l - :E v I Tv (x) I Avl:,.

n 1.k+1 v=n-lln+2 n .

114]~.

( )l''llfk

vllog,. (v~l)JBH-112 ~J

[(ii) 1 l n · + E -- }:; Av l Tv (x) I n A"+l v=n--ln+2

n

l k]lfk)k ( ):" p.u - - - J , = Nlfk.+ N~fk. ,say.

We oQserve that

n (ii) .·• r '. N1 = O(l) E ~

n i\ · E vlTv(x))"i\: 6

v=n-An+2 n

t v[log/iv:6J B+s-1/2 l co

= O;•~l). E. v ! Tv(x);jk.i\~'. 6 v=l

I v[log"(v:i·IT lh11-112}.

(ii) l E-

n;;.,v Ak+l n

0o f. ··· µv l = 0(1) E v j Tv(X)l1!! A. \{log,.(v + !) ]lh-111161 ~ v=l

= 0 (l),by(J.2).

And similarly

(ii) __ I··-.. E: N2 = O(l) E; Ak+i v=n-Ani-2 n. n

I T. (x) jk )"-1 .v i•v.. ·/J.v ..

(Iog"'(v t-l)Bt-&~112

[115 '

00

= 0 {1) ~ .. I Tv (x) Jk f'v v= 1 A,; [logio (v+l) Jli~,,:,-~1-';2--· :::;;; 0 (l,), by (3. J).

Therefore

(ii) A~-1 l M.-.1 lk = 0 (1). 1':

fl

Finally.,

(ii) ~ 1.:~1 1 M~ lk +

(ii) ~ 11~~1 'J M? lk

n n

GO = 0 E I Tn.(x) lk p.n: .

n= 1 A11[logi-(n+ l)]li+s:-1/2 ~·· 0 (l), by (3, 3). ·

4. Proof of Theorem 2. To prove this theorem it is suffici~nt

to calcu!llte the order of the integral

I= l: f(w) { :w P~~.i (cos w)} (sin w)2A-I dw

under the hypothesis of the theorem.

We have

I= J: ¢ (w) ld~ P~~>l (cos w)Jdw

[ ~ = 211 sin w P:/+i1 __ (cos w) ct>1 (w) Jo

- 2A I: w1 (w) d~t~in w p~>.+t> (cos w) t~w = / 1 - 12 , say. · ·

116j.

But,

Ii = 0 { n2A+1 ·IJ ·r;2+1h< (log,. 1/'IJP'11, } = 0 { (log,. n)i /7'}• . .

/ 2 = 2A ·1·1J _:!__ ~ sin w P~>..+1 > (cos w) }. 0 dw l

{ r(l~~) J: (w - ur°' ID,. (u) du } dw.

2A ('iJ rJ'YJ d . · r 0 -~) J 0 ©a. (u) l U dw {Sill W p~A+l) (COS w)}

(w - ur• dw J du = ~pin+ J.'YJ ... }. cl>,. (u) F (·IJ· u)du . · lJ o 1/n

= /2, 1 + lz, 2• say.

We approximate F('YJ, u) for the first integral' T2, 1 i e , when u is

ranging from 0 to l/n

F(", u) ~ l: (w--ui-• d~ 1 ;in w P '''"(cos w) } dw

~ r 2u . i·IJ } { } = l Ju + 2

u (w-u)-" -d~- sin w p~A+I> (cos w) dw

2u · = K I u (w-u)-°' f cos w p~A+I> (cos w)

+ sin w d~ [P:'+>'(cos w) ]}dw +

+ 0 u-"' J~ dd [ s~n w. p~l+I> (cos w; Jd~ 2u ~ ~ ~ 'YJ 2u w ·

[117

;...... Q (n2H 1) 1'1-oc + 0 (n2i+3) u3-rz. + 0 ,u-rz !·sin w p~A+ll( cos w) r ]~ . L. 2u

= O (t12A+l) u1-rz + 0 (n2A+3) u3-o. + 0 (112A+l-L>) u-rz.

Again we have the following order estimate for F('IJ, u) when u ranges

from I/n to 'Yi·

F(ri,u) = [ Jll+(l/n) + I" 1 (w-up

u u+(l/n)

d~ {sin w P~"+1 > (cos w) }dw=~n..\)11..-1 u~..\:...1+u2-A.-2 nA+l O(n~~l) +

+ O(nrz) max [sin w P~>..+1> (cos ,w) J ; . u+(l/n) <;~.;;;;'fJ u+(l/n)

= O (nA+..-1) u->..-1 + O (nA+"') u-..\ + O(n°'+A) u-J.-i.

If u + (l/n) is not less than "IJ we do not need to break the integ

ral il'lto two parts.

Now

[1/n 12, l = JO @ (u2+P-°') (log"' l/u)Y fk n2A+1 du +

[Ifn + Jo 0 tu4+P-°') (log"' l/u)Ylfl n2H3 du

Il/n . · + .

0 ~ (ui+P-a.) (log"' l/u)Y/k n21+1-11 du

1'l8j

== o ~f i'"'t3+ih•>+2>.'.¥1 (log,.. ·n}'r 1i} + d{ ti-<1>+lf-"ol>+s~ 2,\ (log,.. np 11c}

+ O·{n-<2+lhii1+2.Hl-A_(logi> n)J\/k}

= o {n-<2+1h•l +2,\+l (log11- n)Ylk} +

+ 0 { n2Hl-A(3+1l-0tHAl2+1hn-<2+1l-0t>.(fog ... n)Yfk}

-::-: O {(logi>.n)YI"},

and

12, ~ L ~ 0 [nH°'""l] J:/n ull->. (log,.. 1/u)'!o.n, du + .

. f 0 [n"~°'i] J;/n ~HP-'A {log,.. lfu)Y/1' du

+ 0 [n~t-,\-A ·r"fl . u1+1l-,\~l [logi> 1/u]Yl7'

.. 1/n ... · .. .·

= o {nl:1-rt-t.<I+ll-,\> (log11- n)Yfk} +

+ o {n2>.+t-A<3+1l-0t1-11+A-0t>+A<lH.-0t1 (1og11-n)Y/k}

+ O {n~~+i,-4<3t~-0t:i ,...1..,,~+~·;-"Ahl+"Ot-"k> :(tug11- n)Y.'k}

= 0 { (log11-n)Yl1'},.

and therefore

I = Q { (log""n)Y 1k}

Now, using ste-ps parallel to those used in the ttoof of Theorem 1, we see that the proof is complete•

[119

REFERENCES

[I] B. K. Beohar, On absolute Cesaro summability of Laplace series,

Riv. Mat. Univ. Parma (2) 8 (1967), 197-203

£2] R. K. Jain, Ashok Ganguly and B. K. Madan, On I 11,.\ j 1t

summability factors of Fourier series, Indian J. Pure Appl.

Math. 9 (1978). 282-·289.

[3] E. Kogbetliantz, Recherches sur la sommabilite des series ultra

spheriques par la methode des moyennes arithmetiques,

J. Math. (9) 3 ( 1924), 107-187.

[4] G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq.

Publ., Vol. 23, Third edition, Amer Math. Soc. Providence

R. I., 1967.

[5] S. P. Yada,v, On l C, 1 j -Summability of Jacobi series, Indian

J. Pure Appl. Math. 8 (1977), 538-545.

[6] S. P. Yadav, On I C, 1 I -summability of ultraspherical series.

Indian J. Pure Appl. Math. 10 (1979), 1076-1081.

Jiianabha, Vol 15, l 985

ON STRONG APPROXIMATION TO A FUNCTION BY ITS

JACOBI SERIES

By

W. T .. SULAIMAN

Department of Mathematics, College of Education,

University ~f Mosul, Mosul, Iraq

Depart:ment of Mathematics, Heriot.::Watt University,

Edinburgh EH14 4AS, U. K.

(Received; Dec0

ember 21, 1982;R~vised rfinal}: N~vember JS, 1984)

1. DEFINITIONS

An infinite series Eun is said to be strongly summable with in.dex

k > 0 to the sum S, if

n ~ l S, - s jk = 0 (n), n -t"OO

r=O

where Sr denotes the rtn partial sum of the series 'i.:~n. ;

Let f (x) be a function defined on theinteyval -1 < x <: 1, $Uc;4

that the integral

[~1 - xr• (1 +_ x)~ J (x) dx

exists in the sense of Lebesgue. The Fourier-Jacobi expansion, gener

ally known as Jacob~ series corresponding to the function /(x) given by

00

f(x) "' E an P,/tJ:.'3) (x) n=O

lLl)

.·,

1221 ,

where p.,,co.'I>> (x) denotes the Jacobi poJynomial of order (a, {3) and

degr~e ,n iµ x,

and

an= _ _!__ J1

(1--x)" (l+x)ll f(x) p,.<o.'l?>l (x) g'll -1

gn ~ 2io~-p'.~1

2lt4-ia.;·:!4hhil ;_,-.\

r(~+~+1)' r(n+~+l) ~r{ti+1h) n~+.q;+f?i+:1 J

· y1ttl:lsphericaland Legendre series are particular cases of the series

(1;1)-when oc '-:-~ ,,;·• ~ ....;·! a~d a -- 13· = 0, respeetively.

We write . 1~·= ' ·',;

tf(w) =/(cos 11') -f(l),

and establish the following result. ·

Theorem. If a> - 1, {3 >.....;.I, oc + {3 > 0, and if'¥ (t) is a

positive inc~~~fi~tri1.fl!~Mo11;.~~<;f1, th~t

;:t .-

\

t ., , , , , , , , '~, , ; . . Y ', ''jd¢(u{i ~'A'F"tf), ·A' being a

. 0 , ·,:·'t-'" ,·

con's'tant,

t--+0

I: wo-.-1/2 'Y( w) dw = o\:o.+J,/2 iy (t)}, (I.2)

and

·· ... IP (t) = {Wwli+112 ¢(7t-w) dw"-;. 0 {t:/~~-,z:o/(t)}, lt , , . (I. 3)

t~o

then.

11

-~_:_ 2: - ISr«(l~;- J('lYI = :Oftr:(1/n)}. n+l r=O

2. LEMMAS

[123

We require the following lemplas for the pmof of the theorem.

~e~~a.11< (S~go [~l). ·Let rJ;; frbeiarbitrary real numbers and C

a fixed positive ca~stant Then, fa'r ~.:+· oo •

. __ ._{0~;i;..i112. d{~-ct/~), .• -C/;ti-~ e ' 11;_ f2 pf"'-• 13> (cos 6) = .. • -- ·.: - ')- ~: -

" O(n°') , 0 tr;, 6 <;, Cf n

.. ;

Le~ma -~ (Szego f~J). If p. > --·t ~ :> -1, then . '.; ,i--'

p 10.;l'l>·(eos 0) "

n-l12K(6r·{cos (N6 it- fy) + (n sin er1. O(l)}

.Cf n ~ fJ ~ ..,_ C/n where

K(&) = tt-li2(sin 6/2r"'-112.(f<>s 6/2')-tia-112 ' -

N = n + Ha+~ + 1), y = -- (a +- -k) 11;/2.

Lemma 3. The condition(l.3) irripl-;(?$; .

J.t w21i I cfo(tt: lf), .. J dw, =;= O{f"+131f (t).}. 0 ' · __ - - ' ' ·''

(2.1)

Proof of Lemma 3 is fairly straightforward, and we orqJtt.b.e ... ~taif s. -~ /.•,; ·, ~·' .'i"-. ·' ; I •

3. PROOF OF THE THEO.RE.M '' - - • ,. ;. • ! ·.-. ? ~

Lets,. be the ntn. partial sum ofthe series (l. l). Hence we have

Sn {t) =;o . "I·i. p' ~ cos ~)°'~~.If~ ( l :,\- cos w)ll+If2/(cos w). 0 '

124j

~ _ _!_ P!:' Ii> (cos w) P;:' ii> (l) dw m=O gm

S,,(l) -f(l) = I: _(sin w/2)2°'+1(cos w/2)21Hl ¢(w),

. F(n + ti«.+ 13 + 2) p~o.+u1<11'"(cos w) dw r(a + 1) r(n + (3 + 1)

r(n,,+ a + (3 + 2) r(a -f 1) r(n + @ + I) •

r · 1 J I l/n 18 · .. _ .. ·1~--l/n . • 1-~-· . · 1

--~ + + + . }-· ... I I . L 0 ... l/n. · .. o.·. ~:-:J/n)

_; · =J1+12 + J3 + J4, say.

Ji = 0 (n2ao+2) J l/n ~~°'+11 ¢ (w) I dw

. 0

As ef; (0) = 0, we see that

I if> (w) I ~ :Jw d I q, (u) I ·-~ Aqi- (w).

0 . : '···• .....

there'fore ; -~ I· , ,~ ;, c • >

J1 = O (n2ao+2) 1 l/f1w20t+l ~{~) •dw "'" 0 {'Y Oinh. . o.

, ~,.,:: ' ,. ;!" • : .- ; ., • '

J8 . .

J2 .=: O (n00 +1) . (sin_ .y/~)2ao_+.~ fo>s .w/2)213+1 I if> (w) I [Ofo-112).

1/n . ·

124j

~ _ _!_ P:,.°'' Ii> (cos w) P;:' Ii> (I) dw m=O gm

S,.(l) -f(l) = I: _(sin w/2)2<><+l(cos w/2)2lhl rf>(w),

. . . . F(n + tx.+ 13 + 2) p<o.+blif~cos w) dw . F{a + 1) F(n + (3 + 1) n

r(n,+ a+··~+ 2) r(a + 1) r(n + (l + I) •

f . . l Jll/n .··18 . 11(-lfn -~ 1-~-· . ·1 ~ + + ; }-l I

_LO .. _ I/n · .. _.a. 'r':-:1/nj

.· =J1 + !2 + la + J4 , say.

J 1 := Q (n2"'+2) J l/n ~2~+1 j rp (w) I dw

0

As .p (0) = 0, we see that

I rf> (w) I < <jw d I .P (u) I <A-qi (w).

0 .

thete'fore '1" •.• ' i "-· · :. •• >; · ·

Ji= O (n2<11+2) J l/nw20.1-1 4-(~)dw =' b {'F (1/n». . 0

- ",. .!' -· >' :·· • •

.. ~ l < '

Ja . .

": J2.::. 0 (n<11+1) . . (sin; w/~)2<11+.~ (cos ,w/2)2P+l '· p (w) I [Ofo-112).

l/n

n2s

• {sin w/2rr3/2. (cos w/2ra-1i2 (0(1) + n(sin wrl)}]

\3 ... ' . J(; = o (n°'~i1!!} j wi'+i,1~ j1 \fi(w) l «M 4" V(n"'-112) w«""s!2i tfo(w)ldw

1/n l/n

= 0 (n<><+l/2) r wrl/2 'Y(w)dw

1/n

= 0 {'f"(l/n)} by (1.2)

rc-1/n

J3 = O(n°'+l/2) 13

(sin w/~)°'-ll~(cos w/2)PH/2 j rfo (w) I dw

n-l;

(3.2)

= O (n°'+l/2) J (cos w/2)"--l/2(sin w/2)P+i 12,,1 cfo (rc-w) I dw

i/n

=. 0 (n"'+l/2). 0 (l/n)'"+1 12,'f" (l/µ,), b~ (13)

·~~ l·v t\ir (r111·)}. (3.3)

J4 = An r (sin w/2)2°'7.~.~99~ w/2)2P+l I rfo (w) I Pn(O<tl'P> (cos w)dw

rc-1~ · ·

where A,. r'(n '+ a +·~ + '2) .

r(ix + l)T(n + {) + I) •

i ritttreref.ore

. "'~1. .. . ... . J6 =· (- 1)" A1!J1 n (cos w/2)2~+l(sin w/2)2il+1 rfo('Zt-w).

0

1261

.. P .. <P' a.+u cos w) dw.

Sioce Pn<°'' P> (-x) = (-1)" Pn<P' a.1 (x), it follows ther.

f I/n j 4 = 0 (n«+!hl) J w2P+l j </; (rc-w) ! dw

0

0 (n°'+f5) 0 (1/n}'"+a 'Y(l/n), by (2.1)

0 {lf(l/n)}.

In view of (3.1 ), (3.2), (3.3) and (3.4), we get ·

I Sn (l)·- f(l) I =0 {'Y (l/n)}.

Therefore

n 1 ---

1 k /S,(I) - f (I) I = O{'Y(I/n)}

n+ r=O

This completes the proof of the theorem.

(3.4)

For several analogous results, see the works 6f Singh [l] and

Yadav [3].

REFERENCES

[I] T. Singh. Degree of approximation by Cesaro means of Founer

Laguerre series, Indian J. ·Pure App/. Math. 9 ( 1978),

394-399.

[2] G. S;:egC>, Orthogonal Polynomials. Amer. Math. S0c. Colloq.

Publ., Vol. 23, Third edition, Providence, R. L., 1967.

[3] S. P. Yadav, On I C, 1 j -summability of Jacobi series, Indian

J. Pure Appl. Math. 8 (1977), 5::.8-545.

Jiianabha, VoL 15, 1985

A NOTE ON A CLASS OF ANALYTIC FUNCTION$ IN

THE UNIT DISK

By

SHJGEYOSHi OWA* .

Department of Mathematics, Kioki University

Higashi-Osaka, Osaka 577, Japan

(Receii>ed: Marc.h 25, 1984: Revised: December 12, 1984)

ABSTRACT

It is the purpose of the present paper to prove some distortion

inequalities for the derivative of functions belonging to a certain class

of analytic functions considered earlier by K. S. Padmanabhan [3],

and by S. Chandra and P. Singh [1].

1. INTRODUCTION

Let A (a) denote the cla»s of functions f(z) of the form

00

fl..z) = z + ~ a,. Z"

n=2

which are analytic in t:Q.e unit disk U = {z : I z J < 1 } and satisfy

the fdlowing condition

(2) f(z) _ 1 I< oc -~- I

Z I

(z EU)

* This research was completed at the University of Victoria (Victoria,

British Columbia, Canada) while the author was on study leave from

Kinki University.

128]

for some a (O < a ~ I)·

This class A (:x) was sfTiaMa By P~litilftt,bhan [3}, and by Chandra

and Singh [ 1 ].

Now, we need 1he follo\ving lemma which is immediately obtained from Nehari's lemnia {2];

;:.•r.", :, ~' ,,.·<, t ,-;:.;,_.~,., ,,.::' C ~ "'~'::-+ >,~

Lemma, Let cf> (z) be analytic in the unit disk U and satisfy

I cfo(z) I~ ix f~r~ ··~ u. fh~~

(3) I ~ '(z) I ' a

~ 1- i ~(~) 1'1i)~j~ l-lzf2 (z E U),

-,':;~"J'~·:r-· ,f· ;'., .. ;::, .--~"-'" <.. :~ :,t·l: where 0 < a ~ 1:

2. MAI_N RES UL TS

We now state and proy~ ~rnr, r~sults:: which provide upper and

lower bounds for the derivative of functions belonging to the class

A (ri)

Theorem l. Let the function f(z) defined by (1) belong to the class

A (a). Then

.. ·I "-~·

(4) If' (z) I ·. .. . .. ···.·· . ,,~ fz 12

~ i . + 2a I z I + I - l z 12

and

(5) j !' (z) I ~ 1 - 2a; I z j - __ ix I z 12

1 - /Z12"

·Jo~· o "< a ~ t d~'<ri'~ Y;:·

PROOF. Let the function g (z) be defined by • ,

{129

(6) g (z) = f (z) z

- 1.

Then g (z) is analytic in the unit disk U and has simple zero at the

origin. Hence we can write that

(7) g (z) = _j_ (z) - 1 = zrj; (z), z

where <fo(z) is analytic in the unit disk U and satisfies \ cfa(z) I < a for

z E U. Thus we have

(8) f (z) = z + z2<fo (z);

hence further,

(9) f (z) = 1 --!-- 2z<j,(z) + z2cfa (z).

Consequently, by using the lemma, we obtain two inequalities ( 4) and

(5) for 0 < a ~ I and z EU.

Next we shall consider functions with initial zero coefficients.

THEOREM 2. Lr:t the function defir.ed by

00

(10) f (z) = z + ~ Gn+p zn!-p

n= l

belong to the class A(a ). Then

(11) If (z) I ~ 1 + (p + l}x I z\P +

and

(pE N = {l, 2, 3, ... })

aj z /PH

~2

(12) a I z fv+l f.f'(z) I ;;;;, 1 -(p+ 1) a I z/P - 1- 1'z1~

for 0 < °' ~ 1 and z E U.

130]

PROOF. Let the function h (z) be defined by

(13) h(z) = _!_ (z) z- - I.

Then h(z) is analytic in the unit disk U and hasp zeros at the origin.

Therefore, we can write

(14) h(z) = f(z) z

-1 zP<j;(z) ,

where <P (z) is analytic in the unit disk u and satisfies I <f;(z) r < IX for

z E U. Th is gives two inequalities of the theorem with the aid of

Lemma.

3. REMARK

We have not been able to obtain sharp estimates for I f' (z) I in

our theorems.

REFERENCES

[I] S. Chandra and P. Singh, On certain classes of the analytic

functions, Indian J. Pure Appl. Math. 4 (1973), 745-748.

[2] z. Nehari, Co11formal Mapping, McGraw-Hill Book Co.,

New York, 1954.

[3] K. S. Padmanabhan, On the radiu~ of univalence and starlikeness

for certain analytic functions, J. Indian Math. Soc. (N. S.)

29 (1965), 71-80.

Jiianabha, Vol 15, 1985

A NOTE ON THE RADII OF STARLIKENESS AND

CONVEXITY

By

SHIGEYOSHl OWA * Department of Mathematics, Kinki University,

Higashi-Osaka, Osaka 577, Japan

tReceived : April 26, 1984; Revised : February 7, 1985)

1. INTRODUCTION AND DEFINITIONS

Let A cenote the class of functions of the form

(1. 1.) 00

f (z) = z + ~ an zn n=-2

which are analytic in the unit disk U = { z: I z I < l}. We denote by

S the subclass of univalent functions f (z) of A and by S* and K the

subclasses of S whose members are starlike with respect to the origin

and convex in the unit disk U, respectively. A function /(z) in A is

said to be starlike of order rx (0 ~ ot < 1) in the unit disk U if and

only if

(l. 2) Re f _!f_Jz) } f (z) - > ot

for z E U. Further, a functionf(z) in A is said to be convex of order

ix (0 ~ a < 1) in the unit disk U if and only if

*This research was completed at the University of Victoria (Victoria, British Columbia, Canada) while the author was on study leave from Kinkl University.

1.32[

(1.3) Re { 1 + z.f'' (z) l > a f. (z) ~

for z E. U. We denote by S* (ri) and K (a) the subclasses of A whose

members satisfy (1.2) and (1.3), respectively. Then it is well known

that S* (C() C S*, K (C() C Kfor 0 < a < 1 and that S* (0) = S* ,.

K (0) = K for a = 1.

The classes S* (a) and K (cc) were first introduced by Robertson

[9], and were latter studied by Merkes, Robertson and Scott [5], Schild

[12], Bogowski, Jablonski and Stankiewicz [I] and Jack [3]. In

particular, the class S*(l/2) was studied by Schild [13] and

MacGregor [4]

Now, Several essentially equivalent definitions of the fractional

calculus, that if' the fractional cerivatives and the fractional integrals,

have been given in the literature (cf., e.g., [2, Chapter 13], [6], [7], [IOJ

[11], and ll4, p. 28 et seq.]). We find it convenient to restrict ourselves

to the following definitions u~ed recently by Owa [8].

DEFINITION 1. The fractional integral of order ,\ is defined by

(1.4) Dz-Af(z) = I'~,\) J~ ftO _. _ d~, (: - t51-;i.

where ,\ > 0, f (z) is an analytic function in a simply connected reg10n

of the z-plane containing the origin, and the multiplicity of (z - ~);i.-1

is removed by requiring log (z - ~)to be real when (z - ~) > O.

DEFINITION 2. The fractional derivative of order A is defined

by

(1. 5) DzA/(z) 1 d

r(t - A) dz- J

7 - f (~) 0 (z - ~)A

d~,

f133

where b ~ ,\ < 1, j (z) is an analytic function in a simply connected

region of the z-plane containing the origin, and the multiplicity of

(z - ~)-).. is removed by requiring log (z - q to be real when

(z - ~) > 0.

DeHniticn 3. Under the hypotheses of Definition 2, the fractional

derivative of order (n + A) is defined by

dn (1.6) Dzn+A f(z) = ---- D2J.. f(z),

dzn

where 0 :3 ,\ < 1, n E N U {O} and N. = {l, 2, 3, .. }. '

Recently, by using the fractional derivative Dz). f(z) of /(z) of

crder ,\, Srivastava and Owa [ 15] introduced -the following class.

Definition 4. We say that f(z) is in the class PA *(.x, ~) if f(z) of

S defined by

00

(l.7) /(z) = z - ~ a,.zn n=2

safof ies the following condition

I r(2 - A)z>..-1 D,J.. f(z) - 1

(1.S) lrc2 .= A)zA-1 Dz>- f(z) + (1 - 2a) \ < ~

for 0 :3 ,\ < 1, 0 :3 a < l and 0 < ~ :3 1.

2. Radii of starlikeness and convexity

(a,. ;?; 0)

(z EU)

In this section, we determine the radii of starlikeaess and conve

xity of functionsf(z) belonging to the class P). *(a, ~). We first need

the following lemma given by Srivastava and Owa [15].

134]

Lemma. A function f(z) defined by (1. 7) is in the class PA. *(oc, ~)

if and Of!l{' ,if

(2.l) ;;' I'(n + 1) r(2 - ;\)_ (1 +~)an~ 2~ (l _a'\. n = 2 r( n + 1 - ;\)

The result (2. 1) is sharp •

Now we state and prove

Theorem 1. Let the function f(z) drfined by (1. 7) be in the class

P). *(oc, ~) for 0 ~ .\ < 1,0 ~ a < 1 and 0 < ~ ~ 1. Then f(z) is starlik~

0.1 :Wd<:r y (0 :3 y < 1) tn the disk I z.1 < ro. where

(2.2) inf l(l + ~) (1 - y) r(n + I) I'(2 - .\). 11 .. t<n-11

ro = · nEN-{I}. 2~(n - y) (1..,, a) r(n + I-;-.\) ..

with equality for the function f(z) gil'en by

2~ (1 - a) r(n + 1 - .\) zn. (2.3) /(z) = z (1 + [') I'(n + 1) r(2 1\)

Proof. It suffices to show that

;(2.4) }· zf' (z) - I ) < 1 -:- Y f(z)

for\ z I < ro. Now, v.e can observe that

.00

/,

(2.5) ... 1· .. !:_·{ (z) - 1 l , f~z) ' · ~

~ (n - 1) an I z I n~l n=2

00

1 -· ~ an l z 111-:-l n=2

~ 1 - "(

if and only if

(2.6) '; (n· - y ) an I z !"-1 ~ 1 . n =2 1 - Y

[135

Consequently;, in ·view of the lemma, we need only find values of I z 1 f6r which

(2.7) ·(n. - Y) I z 1·.-1 ·=:; (1 + -~}. r(n1+ 1) r(2 - ii) . 1 ,....... Y - 2~( 1 - a) F(n + l - ii) (n ~ 2

)

which will be true when J z I ~ r 0 This completes the proof of the

theorem.

Theorem 2. Let the function f(z) df,jined by ( l. 7) be. in the class

PA. *(a., ~J for 0 2 A < 1, 0 2 a < l and 0 < ~ ~ 1. The~f(z) is convex

of order y (0 ~ y < l)'·i~ the disk '1 z I ~ ri. where

. inf { (1 + ~)(l :.___y)T(n) r(2 - ii) {2.8) 11 = nEN-{1} 2.,ln ___:. Y) (1 - ~)T{n + 1-il)

witi2 equality for thefunc:tion f(z) given by

2~(1 - a.)J:'(n + 1-:-ii.), zn. (2.9) /(z) = z -;-- (T+ -~) ni:'(n + Iyr{2 - ii)

'.;'il

} ll!n-ll

P~oaf. Sine~ f(z) E K (y) if and only if zf'(z) E. S*(y), we can see

that the theorem folloW3 that of Theorem 1. with an replaced by nan.

Corollary. Let the functionj(z) defined by (l.7) be in the class : ' ' ,, ( . ' ·. . . ~ . ; : ·' '

P 1. *(a, ~)for 0 ~- < Ii l, -0 ;§ oc < 1 and 0 < ~ ~ 1. Then f(z) in univ-

ale1!t and, st.arlike fw l z i < r2, where .'.

(2.10) ~2 = . inf { ( l + ~) r(!1) r(2 - ii) __ }· l/<n:ll n E N - { 1 } 2~( l - oc) r( n + 1 .:.... ,\) . · .

1361

Proof. By taking y = O in Theorem 1. we have the above coroll

ary.

Acknowledgements

The author wishes to express his thanks to Professor H. M.

Srivastava, for his kind encouragement and helpful guidance.

REFERENCES

[l] F. Bogowski, F. F. Jablonski and J. Stankiewicz, Subordination

en domaine et inegalities des modules pour certaines classes

de fonctions holomorphes dans le cerc!e unite, Ann. Univ.

Mariae Curie-Sk/odowska. Sect. A 20 (1966), 23-28.

[2] A. Erdelyi, W. Magnus, F. Oberhettinger and F G. Tricomi, .Tables of Integral Transforms, Vol, II, McGraw-Hill Book

Co., New York, Toronto and London, 1954 ..

[3] I. S. Jack, Functions starlike and convex of order ex, J. London

'Math. S~c~ (2) 3·(1971), 469-474~

[4] T. H. MacGregor, The radius of convexity of star.ike functions

of order 1/2, Proc. Amer. Math. Soc. 14 (1963), 71-76.

[S] E. P. Merkes, M. S. Robertson and W. T. Scott, On products

of starlike fur.ctions, Proc. Amer. Math. Soc. 13 (1962),

960-964.

[6] K. Nishimoto, Fractional derivative and integraL. Part l,, J.

College Engrg. Nihon Univ: Ser. 13, ~7. (1_?7?) •. ")i~.i.9:· .'," , [7] T. J. Osler, Leibniz ruls for fractional derivatives generalized

and an applicat.ion to infinite series, SIAM J. Appl. Math.

18 (1970), .. 658 ·674.

[137

[8] S. Owa, On the dist6rtion theorems. I, Kyungpook Math. J. 18

(1978), 53-59.

[9] M. S. Robertson, On the theory of univalent functions, Ann. of

Math. 37 ll9J6), 374-406.

[10] B. Ross, A brief history and exposition of the fundamental

theory of fractional calculus, in Fractional Calculus and Its

Applications (B. Ross. ed.), Springer-Verlag, New York,

Heidelberg and Berlin, 1975, 1-36.

[l l] M. Saigo, A remark on integral operators involving the Gauss

hypergeometric functions, Math, Rep. College General Ed.

Kyushu Univ. 11 (1978), 135-143.

[ l 2] A Schild, On scar like function;> of order ex, Amer. J. Math. 87

(1965), 65-70.

ll3] A. Schild, On a class of univalent, starshaped mappings, Proc.

Amer. Math. Soc. 9 (1958), 751-757.

[14] H. M. Srivastava and R. G. Buschman, Convolution Integral

Equations with Special Function Kernels, John Wiley and

Sons, New York, London, Sydney and Toronto, 1977.

[ 15] H. M. Srivastava and S. Owa, A new class of univalent functions

with negative coefficients (to appear).


SOME REMARKS ON COMMON FIXED POINTS OF

FOUR MAPPINGS

By

M,' L. DlVICCARO and S. SESSA

Unive~sita di Napoli Facolta di Architettura

Istituto di Matematica Via Monteoliveto, 3.

80134 Napoli, Italy

(Received: March 1, 1985 )

ABTTRACT

In this paper, inspired by a recent. result of Fisher [4],

we point out a comqion fixed point theorem for four self mappings of

a complete metric space, using a well known contractive condition of

Meade and Singh [ 13] and the concept of weak commutativity of the

second author [16]. Our theorem generalizes results of Chang [l],

Imdad and Khan [8], Sessa and Fisher [17] and Singh and Singh [19].

1. INTRODUCTION

Let R+ be the set of nonnegative reals and let (X, d) be a complete

n~etric space. Meade and Singh [13], improving a result of Husain and

Sehgal [6], prcved a common fixed point theorem for two selfmapp

ings of (X, d) considering a real function f : (R+)5 -+ R+ satisfying

the following properties :

li) f is upper semi-continuous

AMS (MOS) Subject Classification : Primary 54H25, Secondary

47HIO.

140)

(ii) f is. non-decreasing in each coordinate variable,

(iii) f (t, t, at, ht, t) < f for any t > 0 where a > 0, b ;;?: 0 and

a+ b = 3.

Let F be the family of such functionsf. Many authors studied cont

ractive cohditions with funetions f" Fox using fur.cfions with similar

properties: fot instance, see C. Chang[T], S. Chang [2], Danes [3],

Guay, Singh Whitfield [5], Husain and Sehgal [7]. Imdad and Khan

[8], Imdad, Khan and Sessa'[9]; Kasahara arid Singh [10], Matkowski

[12], Park and Rhoades [14], Rhoadesr[l5], Sessa and Fisher [17},

Sharma [18.], and Yeh [20, 21].

Inspired by a recent paper of Fisher[4], we prove a common fixed

point theorem for four self mappings of (X, d) exten9ing the results of

[l]. [8]. [13], [17], and [18].

Wealso;use the'following·notion~of· weakly· commuting· ~elfmapp

ings of (X, d) given in,[ lq].

:Qefi~i~io,n. Two selfmapp,i11gs tS' and I of (X, d) weakly coin mute if

rf (SJx,JS.1:) ~ d(Jx, Sx)

for any x "X.

Obviously, 1f S commutes with l, then S also weakly-commutes with

J but when S weakly commutes with J, t11en S does not necessarily com

mute with I as is shown in Example '1 below .

. 2. A fix~d point theorem!

As in ln. [5] we put

(141

y(t) =c max { f(t, t, t, t, t), J(t, t, 2t, 0, t),f(t, t, 0, 2t, t)}

for any t > 0 and further, we assume a slight modified version of the

property (iii), i. e.

(iii') y(t) < t for any t > 0.

Now let S, T, I and J be four self mappings of (X, d) such that

(I) T(X) C !(X) and S(X) C J(X)

(2) d(Sx, Ty) .;;;; f( d(!x, Jy), d(fx, Sx), d(!x, Ty),

d(Jy, Sx), d(Jy, Ty))

for all x, yin X, where f satisfies (i), (ii) (iii').

Let x0 be an arbitrary point of X and x1, x2 in X such that Sx0

= Jx1, Tx1 = lx2• This can be done since (l) holds. [n according to

Fisher [4], we can inducti'tely define a sequence

(3) Sxo, Txi. Sx2, Txa ,. .. , Sx2,., Tx211+i, Sx2n-;-2, ...

such that Sx2,. = Jx211+i. Tx2n+i = lx2n+2 for each integer

n e N = { 0, 1, 2, ... }. Employing the method of proof of [13), it is

proved that

Lemma. The sequence (3) is a Cauchy sequence.

See also an analogous res ult in [ 17].

Meade and Singh [13] established the following result :

.THEOREM I. Lets and T two self mappings of (X, d) satisfying

d(Sx, Ty) <,; f(d (x, y), d(x, Sx), d(x, Ty)

d(y, Sx), d(y, Ty))

142]

for all x, yin X, where f" F. Then S and. T have a unique common

fixed point.

Bearing in mind the proofs of the results of [I], [5]. [18], it is not

hard to verify that Theorem 1 holds also under the assumption (iii')

instead of (iii). Analogous consideration holds for the main Theorem

of [8]. Drawing inspiration from Fisher [4], we generalize· Theorem 1

with the following

Theorem 2. Let S,T, I and J are four self mappings of (X,d) sat;sfying

conditions (I) and (2), where f satifies properties (i), (ii), (iii'). If one

of S, T, I and J is continuous and if S and T weakly commute with I

and .! respectively, then S, T, I and J have a unique common fixed

point z. Further, z is the unique common fixed point of S and I and

of TandJ.

Proof. It is similar to that of Fisher and Sessa [ 17].

However we outline the essential steps in order to show where the

weak commutativity plays the key role

By Jeimria, the sequence (3) converges to a point z.

Suppose that I is continuous. Since the sequen;;es

{ Sx2n} = {Jx2n+I} and {Tx2n_ 1} = {/x2n}

converge also to z, we have that the sequence {/Sx2'.} converges to lz. S

being '\\eakly commuting with /, we deduce

d(.Six211, lz).,:;:; dl.5'lx2n, !Sx2n) + d(IS>/2n, lz)

.,:;:; d(lx2rt, Sx2n) + d(ISx.,., lz),

which implies, al! ,,_,..001 that {S/x2n} converges to lz. As in [ 17],using

l[l43

twice (2) and properties (i), lii), (iii') and the fact that {/2x2 .. } conver

ges also to /z, we ascertain /z = Sz = z. Since the range of J contains

the range of S let z be a point in X such that Jz' = z. Then using (2)

we have

d(z, Tz') = d(Sz, Tz) ~

f(O, O~ d(z, Tz'), 0, d(z, Tz')) ~ y (d(z, Tz')),

which implies z = Tz' by property (iii'). Since Tis weakly commuting

with J, we have

d(TJz', JTz') ~ d(Jz', Tzr) = d(z, z) = 0

and then

Jz = JI'z' = TJz' = Tz.

Using again (2) and (iii ), one deduces Tz = Jz = z. Therefore z

is a common fixed point of S, T, ! and J.

Ar;alogous proof can be given if one supposes the continuity of

J instead of /,

Now we suppose the continuity of S. Then the sequence {Slx2,.}

converges to Sz. Since S weakly commutes with I, we have

d(ISx2,., Sz) ~. d(ISx2,., Slx2n) + d(Slx2n, Sz)

~ d(Sx2,., lx2n) + d(Slx2,., Sz)

which implies, as n -'>- oo, that the sequerice.{/Sx,.} converges to Sz. By

(2) and properties (i), (ii), (iii), and observing that {S2x2,.} converges

al.so to S z, one proves that Sz = z. As above, one shows that

Jz = Tz = z. Since the range of J contaias the range of T, let z" be a

144}

point in X such that lz" = z. Using again (2), we have

d(Sz", Z) == d(Sz", Tz) ~

f(d(lz", Jz), d(lz", Sz"), d(lz", Tz), d(Jz, Sz"), d(Jz, Tz)) ~

f(O, d(z, Sz"), 0, d(z, Sz", 0) ~ y (d(z, Sz")),

which implies Sz" = z by property (iii'). Since S weakly commutes

with !, we have

d(S/z", !Sz") ~ d(lz\ Sz") = d(z, z) = 0

and therefore

lz = !Sz" = S!z" = Sz = z.

Thus z is a common fixed point of S, T, I and J and making a

similar prnof, the same conclusion is achieved supposing the contmuity

of T insetead of S.

The uniqueness of z is easily proved.

3. Some remarks.

Remark 1. Assuming 1 = J = identity of X, Theorem 2 becomes

Theorem l.

Remark 2. Recently S. L. Singh and S. P. Singh [19] proved a

common fixed point theorem for three self mappings S, T, I of (X, d)

satisfying the following condition :

(4) d(Sx, Ty) ~ h max { d(Ix, ly), d(Ix, Sx),

1/2 [ d(lx, Ty)+ d(Iy, Sx) ], d(Iy, Ty)}

for all x, y in X, where 0 .~ .h <; 1.

. 145}

C. C. Chang [l] studied the following condition

(5) d(Sx, Ty) < /( d(lx, ly), d(Ix, S,\), d(Ix, Ty),

d(fy, Sx), d(ly, Ty))

for all x, yin X, where f verifies properties (i), (ii), (iii').

Further, the cited authors assume I continuous and commuting

with Sand T. S(X) C l(X), T(X) C J(X)-

Of course (4) is a consequence of (5) assuming

f(ti. t2, t3, t4, t5) = h. max {t1, t2, (t3 + t4)/2, t5}

for any ti. t2 , t3 , t4 , 15 ;> 0. However, (2) becomes (5) for I = J and

moreover our assumptions of Theorem 2 are more general than those

cited.

Remark 3. lmdad and Khan [8] proved a co.nmon fixed point

theorem for three self mappings S, I, J of (X,d) satisfying the following

condition

(6) d(Sx, Sy) < f( d(Ix, Jy), d(lx, Sx), d(lx, Sy),

d(Jy, Sx), d(Jy, Ty))

for all x, y in X, where f e F. These authors assume I and J continu

ous, S commuting with I and J and S(X) C I(X) nJ(X). Clearly (2)

becomes (6) for S = T and therefore our Theorem 2 is a stronger

result.

Example 1. Let X = [O, lj with euclidean metric d and let S, T, I

and J defined by

Sx = x/(x + 2), Tx = x/(x + 3), Ix = x/2, Jx = x/3

1461

for any x in X. As shown in [16], S weakly commutes with I. Since

d(TJx, JTx) = x/(x + 9) - x/(3x + 9) = 2x2f(x + 9). (2x + 9)

::;;;; x2/(3x + 9) = x/3 - xf(x + 3) = d(Jx, Tx)

for any x in X, then T weakly commutes with J but T does not comm

ute with J being T Jx -=!= JTx for any non-zero x in X. Let

f(t1, t2, t3,° t4, t5) = t1/(6 + t1) = g(t1) for all t1, t2. t3, t4, t5 > O. It

is immediately seen that f enjoys properties (i), (ii) and (iii').

Further we have

T(X) = [O, 1/4] C [O, 1/2] = I(X), S(X) = (0, 1/3] = J(X)

and

d(Sx, Ty) = I 3x - 2y I / (x + 2). (v + 3) ::;;;;

::;;;; I 3x-2y l /(6 + ! 3x - 2y I )

= g( d(lx, Jy))

for all x, y in X. Being one of S, T, I and J continuous, then all the

assumptions of Theorem 2 are verified resulting 0 the unique common

fixed point of S, T, I and J

The idea of this example appears in [17].

Remark 4. Fisher and Sessa [17), generalizing the results of [4],

established a common fixed point theorem for four self mappings S, T,

I and J of (X, d) satisfying the following condition

(7) d(Sx, Ty) ::;;;; cp ( max { d(lx, Jy), d(lx, Sx), d(Jy, Ty) })

for all x, yin X, where ,p : R+ - R+ satisfies properties (i), (ii) and

. [147

(iii") rfo(t) < t for any t > 0.

Obviously, by putting f(t1, 12, 13, t4, t5) = 0, the condition (2) becomes (7). Now we give

an examle showing that our Theorem 2 is a m6re general than result

of [17], even if one supposes I= J = identity of X.

Example 2. Let X = {A, B, C, D, E} the subset of R2, where

A =: (- 1, 0), B = (0, 0), C =: (0, 1/2), D = (0, l), E =: (-1, 1),

with euclidean metric d. Let Sand T tv. o self mappings of X defined as

SA =SB= SC= SD = C, SE= D and TA = B,

1B =TC= TD= C, TE= D.

Then it is not hard to verify that condition (2) is satisfied if we

choose

f(tl> t2. £3, f4. t5) = (2t1/5 + t2/6 (t3 + t~)/5

for ally t 1 , t 2 , ta, t4, t5 ;> 0. Condition (7) does not hold otherwise if

there exists a function satisying properties (i), (ii), (iii'), we sho.uld

obtain for x = E and y = A :

d(SE, TA)= d(D, B) = 1 :::;;;; <fo (max{ d(A, E), d(E, D), d(A, B)})=

cp (1, 1, 1),

a contradiction to the required condition (iii").

The idea of this example appears in [11).

REFERENCES

[1] Cheng Chun Chang, On a fixed point theorem of contractive type,

Comm. Math. Univ. St. Paul. 32 (1983), 15-19.

148]

[2] Shih-Sen Chang, A common fixed point theorem for commuting

mappings. Proc. Amer. Math. Soc. 83 (1981). 645-652.

[3] J. Danes, Two fixed point theorems in topological and metric

spaces, Bull. Austral. Math. Soc. 14 (1976), 259-265.

[4] B. Fisher, Common fixed points of four mappings, Bull. Inst.

Math. Acad. Sinica 11 ( i983), 103-113.

[5] M. D. Guay, K. L. Singh and J. H. M. Whitfield. Common fixed

points for set-valr.ed mappings, Bull. Acad. Po/on. Sci. Ser.

Sci. Math. Astronom. Phys. 30 {1982), 545-551.

[6] S. A. Husain and V. M. Sehgal, On common fixed points for a

family of mappings, Bull. Austral. Muth. Soc. 13 ( 1975),

261-267.

[7] S. A. Husain and V. M. Sehgal, A fixed point theorem with

a functional inequality, Pub!. Inst Math. 21 (35) (197 7),

l.19-91.

[8] M. Imdad and M. S. Khan, Fixed point theorems !er a class of

mappings, l1dian J. Pure. Appl. Math. H (1983), 1220-

1227.

[9] M. Imdad, M. S. Khan and S, Sessa, On common fixed points in

uniformly convex Banach spaces, Math. Notae, to appear.

[10] S. Kasahara &nd S. L. Singh, On some recent results on conimon

fixed pofots Indian J. Pure Appl. Math. 13 (1982), 757-761.

[1 i] M. S. Khan, S. Sessa and M. Swaleh, Fixed point theorems by

altering distances between the points, Bull. Austral. Math.

Soc .. 30 (1984), 1-9.

(10

[12] J. Matkowski, Fixed point theorems for the mappings with a

contractive iterate at a point,Proc.Amer.Math.Soc. 2 (1977),

344-348.

[13] B. A. Meade and S. P. Singh, On common fixed point theorems,

Bull. Austral. Math. Soc 16 (1977), 49-53.

[14] S. Park and B. E. Rhoades, Some general fixed point theorems,

Acta Sci. Math. 42 (1980), 299-304.

[15] B. E. Rhoades, Contractive definitions revisited, Topological

Methods in Noni inear Functional Analysis, Contemporary

Math. AMS 21 (1983), 189-205.

[16] Sessa, On a weak commutativity condition of mappings in fixed

point considerations, Publ. Inst. Math. 32 ( 46) (1982),

149-153.

[17] S. Sessa and B. Fisher, Common fixed points of weakly commu

ting mappings, submitted.

[18] A. K. Sharma, Common fixed points of set-valued maps, Bull.

A.cad Polan. Sci. Ser. Sci. Math. Astronom. Phys. 27 (1979),

407-412.

[19] S. L. Singh and S. P. Singh, A fixed point theorem, Indian J.

Pure Appl. Math. 11 (1980) 1584-1586.

[20] Cheh-Chih Yeh, A fixed point theorem in orbitally complete

metric spaces, Pub!. Inst. Math. 24 (38) (1978), 197-199.

[21] Cheh-Chih Yeh, Some fixed point theorems in complete metric

spaces, Math. Japon. 23 (1978), 27-31.


SOME FIXED POINT THEOREMS FOR PAIRS OF MAPPINGS

By

B. E. RHOADES

Department of Mathematics, Indiana University,

Bloomington, Indiana 47405, U. S. A.

(Received : March 7, 1985)

In a recent paper [1] fixed point theorems were established for

certain expansion mappings. An examination of the inequalities used

in both [ 1] and this paper discloses the fact that further generalizations

are doubtful. This paper establishes some fixed polnt theorems for

pairs of mappings.

Theorem 1· Let f. g be surjective selfmaps of a complete metric

space (X, d). Suppose there exists a constant a > 1 such that

(1) d(fx, gy) "> ad(x, y)

for each x, y in X. Then f and g have a unique common fixed point.

Proof. let x0 E X. Since f is surjective there exists a point x1 E J-lxo.

Since g is surjective there exists a point x2 E g-tx1. Continuing in this

manner o:r:e obtains a sequence{x .. }with x2a-n E 1-1x2n, X2n+2=g-lx2n+i·

Suppose x2,.+1 =X2n for some n. Since d(x2n+1' X2n)=d(gx2n+2• fX2n+i),

from (1), d(x2n+2, X2n+1) = 0. The condition X2n+i = x2,. implies that.

x2,. is a fixed ponit off Since also x2,<+2 = x2'11+l• x2n is a fixed point

of g. Similarly, x 2n+2 = x2'11+1 leads to x2'11+1 being a common fixed

point of fand g.

Assume x'll =j::. x'll+l for each. n. From (1), d(x2'11, X2n+i)

152]

d(fX2n+1 , gX2n+2) > ad(X2n+i , X2n-f.2) and d(X2n+1 , X2n+2) =

d(gx211+2 , f'\2n+3) > ad(X2n+2 , X2n+3)~ Therefore d(xn, Xn+i) ):

ad(xn+J> x2n+2), which implies {x,,} converges to a point x e X. Let

ye J-Ix. The assumption x,.-::/= Xn+i for each n implies that x,.-::/= x

for almost all n, From (I),

d(X2n+i, x) = d(gxn+2• fy) > ad(x2n+2, y).

Taking the limit as n-+ oo yields x = fx. Letze g-ix. Then

d(X2n, X) = d(fX2n+l, gz) > ad(X2n+J, Z),

and we obtain z = x, and xis a common fixed point off and g.

Condition ( l) forces uniqueness of the fixed point.

Theorem 2. Letf, g be surjective selfmaps of a complete metric

space (X, d). Suppose there exist nonnegative functions p, q, r, s, t,

satisfying

(2) inf X t'p(x, y) + q(x, y) + t(x, y)) > 1, X, J' E

(3) x, ;n:x {(I - q (x, y) + r(x, y)), (1 - p(x, y) + s(x, y))}> 0,

sup .(4) x, Ye X { p(x, y), q(x, y))} < 1,

and

(5) d(fx, gy) ): p(x, y) d(x,fx) + q(x, y) d(y, gy) + r(x, y) d(x, gy)

+ s(x, y) d(y, fx) + t(x, y) d(x, y)

for all x, y E X, x -::/= y. Then f and g have common fixed points.

Proof. Define {xn} as in Theorem I. Suppose X2n = X2n+1 for some n.

(153

If X2n+I =I= x2n 1-2 , then from (5),

d(x2n, X2n+1) ~ d(fX2n+i, gx211+2)

"> [d(x2n+l, X2n) + qd(X2n+2 1 X211+1)

+ sd(X2n+2 , X2n) + td(X2n+i , X2n+2),

where p, q, r, s, and tare evaluated at (x2n+I , x2,.+2).

Thus

0 "> (q + S + t) d(X2n+l ' X2n+2)•

If q + s + t = 0 then q + t = 0 which, since p < 1, contradicts (2).

Therefore x 2n+1 = x2Q+2 and X2n is a common fixed point off and g.

Similarly, x 2,.+1 = x2,.+2 for some n leads to x211+i being a common

fixed point off and g.

Assume x,. =I= Xn+I for each n. From (5),

d(X2n, X2n+1) = d( fx2n+l , gx2n+2)

"> pd(X2n+I , X2r.) + qd(X211+2 , X2n+i)

+ sd(X2n+2 , X211) + td(X2n+I , X2t1+2),

Ol

(6) (1-p+s) d(X2n, X211+i) > (q + S + t) d(X2n+l , X2t1+2),

where p, q, r, s, and t are evaluated at (x211+i , x 2,.+2).

Again from (5),

d(X211+1 , X2t1+2) = d(gX2t1+2 , f X2t1+3)

> p'd(x2n+3 'X2t1+2) + q'd(X2t1+2. X2t1+1)

154J

+ r' d(X211+3 , X211+i) + t' d(Xzn+2 , X2n ~3),

or

(7) (1 - q' + r') d(X2n+1 , X2n+2) "> (p' + r' + t ') d(X2n+2, X211+3),

where p', q', r ', s and ( are evaluated at (x2,.+3, X2n., 2).

Inequalities (6) and (7), along with conditions (2) and (3), imply

that {x,.} is Cauchy, hence convergent to some x in X ..

Without loss of generality we may assume that x,. =/= X for infin

itely many n since, otherwise, f and g have a common fixed point. If

there exists an infinite number of integers n such that x2,. =/= x, define

y € g-Ix. Then, from (5),

d(X2n, x) == d( /X2n+i , gy)

;;.:;. pd(X2n+I , X2n) + qd(y, gy) + rd(x2n+1 , gy)

+ sd(y, X2n) + td(x2n+i , y),

where p, q, r, sand tare evaluated at (x2,.+i. y). The above inequality

implies that

d(x2n, x) ;;_:;. (q + s + t) min { d(y, x), d(y, X2n), d(X2n+J, y) }

;;_:;. inf X (q+s+t) min {d(y,x), d~y, x,.),d(x2n+i.Y)}. x, y €

Taking the limit as n _,.. oo yields

inf ( ( 0 > x x q + s + t) d x, y), 'y €

which implies that either x = y or inf X (q + s + t) = 0. How-x, y €

ever, the latter condition, along with (4), contradicts (2). Therefore

x = y.

(155

If x2n+I =F x for all n sufficently large, then x2,. = f'l;2n+i = x.

Taking the limit as n -+ oo yield x as a fixed point off.

In x2,.1 1 =F x infinitely many n, define 1-1x.

Then, from (5), with p, q, r, s, and t evaluated (z,x2n+2)

d(x2n+-I, x) = d(x2n+2,fz)

> pd(z, fz) + qd(x2n+2 , X2n+1) + rd(z, X2n+i)

+ sd(x2n+2, x) + td(z, X2n+2)

> inf X (p + r + t)min{d(z,x), d(z,x2n H), d(z,x2n+2)}. X,J E

Taking the limit as n-+ oo yields 0 > inf X (p+r+t) d(x, z), X, J E

which, in light of (4) and (2), implies x = z.

Theorem 3· Let f, g be surjective continuous selfmaps of a complete

metric space X. If there exists a real number a > 1 such that

(1) d(fx, gy) >a min { d(x, fx), d(y, gy), d(x,y)}

for each x,y e X, thenf or g has a fixed point or f and g have a common

fixed point.

Proof. Define {x .. } as ia Theorem 1. If x,. = Xn+I for any n, thenf

or g has a fixed point.

Assume Xn =F Xn+i for each n. From (8),

d(X2n , X2n+i) = d(f x2n+I, gx2n+2) > > a min { d(X2n+I , X2n) , d(X2n+2 , X2n+i) } •

and

156j

d(X2n+i , X2n+2) = d(gX2n+2 , fx2n+s) ;>

;> a min { d(x2n+3 , X2n+2), d(x2n i·2, X2n+1)} •

Thus, for each n, d(x,., Xn+1) ;>a min { d(xn, Xn+1), d(xn+v Xn+2) },

which, since a > 1, implies that {xn} is Cauchy, hence convergent, to

some x in X. The condition x2n = fx2n+1 , X2n+I = gx2n+2 and the

continuity of fand g imply thats is a common fixed point off and g.

REMARK 1. Setting f =' g in Theorem I and 3 yields Theorems I

and 3 of [1]

REMARK 2, Setting/= g, s = t = O,p, q, rconstantsin Theorem 2

yields Theorem 2 of [l].

REFERENCE

[l) S. z. Wang, B. Yi, Z. M. Gao and K. Iseki, Some fixed point

theorems on expansion mappings, Math. Japan. 29 (1984),

631-636.

Jnanabha, Vol. 15, 1985

EFFECT OF VISCOSITY ON RAYLEIGH-TAYLOR INSTABILITY

IN THE PRESENCE OF A VERTICAL MAGNETIC FIELD

By

B. M. ~HARMA

Department of Mathematics, S. K. N. Agrifulture College,

Sukbadia University, Jobner-303329, India

tReceived: April 5, 1984)

ABSTRACT

The character of equilibrium of a heavy, viscous, incompressible,

finitely condducting and rotating fluid of variable density in the

presence of a vertical magnetic field is investigated, when the lower

bounding surface is rigid and the upper is free. The nature of the

boundaries chosen alter the order of the dispersion relation compared

to the cases where the boundaries are both free or both rigid. It is

found that the stability criterion is independent of. the effects of

viscosity, finite resistivity and rotation. It is further '.}nvestigated that

the growth rates both increase or decrease with the increase in viscosity.

1. INTRODUCTION

The character of an incompressible fluid of variable density stratified

in the vertical direction was first inve;;tigated by Rayleigh [3] and he

found that the configuration is stable or unstable ac~ording as dp dz

is everywhere negative or anywhere positive. Chandrasekher [ l] introd-

uced the viscosity in the Rayleigh-Taylor instablity and observed that

in the stable case, the fluid oscillates about the mean position with an

amplitude which decays exponentially at a rate which increases with

increasing viscosity.

158)

The hydromagnetic version of the Rayleigh-Taylor instability was

further studied by Hide [2] for a fluid having exponentially varying

density in the vertical direction. He included the effects of viscosity

and finite resistivity in his problem. Sharma and Ariel (4] investigated

the effect of finite resistivity of the medium on the equilibrium of a

heavy, viscous, incompressible, rotating· fluid of variable density in

the presence of a vertical magnetic field. The present note investigates

theoretically the same problem as considered by Sharma and Ariel [4],

when the fluid is confined between two boundaries, the lower bounding

surface being rigid ar.d the upper free. Based on the existence of a

variational principle we obtained the dispersion relation for a fluid

having exponentially varying density. The object of this problem,

under the~e boundary conditions, is to find out to what extent

the imtability of the configuration is affected by changing. the

viscosity of the medium.

2. PERTURBATIONSEQUATIONS

The linearised perturbation equations for the problem under

consideration are (Sharma and Ariel {4], p. 104, eqns. (31), (32), (34)

.and (35))

KH gk2 n[k 2 po w - D(poDw) + - 0(D2-k2)Dhz - -(Dpo)w -

4'1C n

- D(2p0 .Q ~) + µ.o(D2 - k2)2 w + '1.Dµ.o (D2 - k2)Dw +2

+ D2µ 0(D2 + k2)w = 0, (!)

KH [npu - µ 0(D2 - k2) - D µt,DJ t; - 4rc 0 DE, 2p00 Dw, (2)

[n -. "fJ (D2 - k2j]hz = Ho Dw; (3)

and [n..,... 7l (D2 - k2)] E, ~:. HoDl.. (4)

where

(J59

Po denotes the density before the system is disturbed, --+

w denotes the z-component of the velocity u at a fixed point, --+

g is the acceleration due to gravity, components (0, 0, -g),

µo is the coefficient of viscosity, assumed to be variable in the undis

turbed state,

K is the coefficieat of magnetic permeability assumed to be constant, --+ Ho is the magnetic field directed towards the z-axis in the undisturbed

state,

n is the rate at which the system departs from equilibrium, his the

z-component of the magnetic field in the perturbed state,

k is the total wavenumber o1 the initial disturbance,

--+ ?;; is the z-component of the vector curl u ,

--+ ~is the z-component of the vector curl h ,

D = d~ and ·IJ = [4rcKpr1.

3. BOUNDARY CONDITIONS

The fluid is assumed to be confined to the planes z = 0 which is

rigid and perfectly conducting and z = d. which is free boundary.

The appropriate boundary conditions for the present problem are

(i) w(O)

(ii) w(d)

Dw(O) = h(O) = ~(O) = DC,(O) = 0,

D2w(d) = Dh(d) = D?;;(O) = C,(d) = 0.

(5)

(6)

Multiplying the equation (1) for the characteristic value nt by w;

and integrating across the vertical extent of the fluid we obtain on

combining equations (2),(3)and(4),after a series of integrations by parts

1601

n(li + h + /5 + /7 + /9) + la + ho -

gk2 - -- 12 + "tjk?(h + 2!5 + 16 + [9 + 110) = 0, (7)

n

the integrated parts vanish on account of boundary conditions,

where

Ii = J

d .

0 p0 [k2w2 + (Dw)2 J dz, (8)

/2 = J: Dp0w2 dz, (9)

f d . f d J3 = µo[k2w2 + 2k2 (Dw)2 + (D2w)2] + k2 D2µ0 dz,

-0 0 (IO)

_ Kk2 f d h2 dz, 14 - 4n; 0 (11)

K Id / 5 = --- (Dh)2 dz, 4n; ~/ . .

{12)

1 - K Jd ~ - ·~J:;-J2 . (D2h)2 dz O' ~ (13)

·rd /7 = O Po~2 dz~ (14)

18 = J: µ 0 [1c21;2 + (DIJ :r1z~ (15)

f9= K fd 4nf2 . ~ 2 dz . . Q,

(16)

[161

/10 = K !d 4~k2 (D~)2 dz 0 '

(17)

4. THE CASE OF EXPONENTIALLY VARYING DENSITY

A case for which a simple analytical solution can be found is one

in which the undisturbed densjty distribution is given by

po(z) = pexp ~z, (18)

where p and~ are consants.

Assuming v, the coefficient of kinematic viscosity to be constant,

we shall take

p.o(z) = vp exp ~z. (19)

In order to ensure that the density variation within the fluid is

small compared to the average density, we make an assumption that

j~dl<<l. (20)

Let us assume th~ following trial functions which satisfy the bou

ndary conditions (5) and (6) :

11 (z) = W(coslz - cos3/z), h(z) = X1 sin/z + Xs sin3/z, ~(z) = Z1 sin/z + Zs sin3/z, ~(z) = K1 coslz = K3 cos3/z,

where I -:;= (~s/2d), s being an odd integer.

l ~ l j

(21)

Substituting the values of w(z), h(x), ~(z) and ~(z) equations (2),

(3) and (4), we get

kHol [n + 1J (/2 + k2) ]Z1 + -- K1 = - 2nlW, 4~p

(22)

162]

[n + ri (912 + k2) ]Zs+ 3 Hol Ks= 60.ZW, 4rcp

[ n + ri (/2 - k2) ]X1 = - lfloW,

[ n + ri (9/2 + k2) ]Ks= - 3/H0 W,

[ n + ri (1 2 + k2) ]K1 = /HoZ1,

[ n + ri (912 + k2) ]Ks = 3/HoZs.

(23)

(24)

(25)

(26)

(27)

Evaluating the integrals in equation (7) with the assumed form of

w .• etc, and eliminating the constants from (22)-(27) we obtain the

following dispersion relation between n and k:

n2 (k2 + 5/2) - g~k2 + nv (k4 + 10/2k2 + 4114) +

n/2 v2 /2 + k2 9(/2 + k2) -2- [ n+ri (/2 + k2) + - n +."fl (912 + k2)

.. + 20.212 n [ __ n + ri (12 + k2) {n + ri ([2 + k2)} {n + 11(12 + k2)} + /2V2 +

+ 9n + 9ri (912 + k2 ) . __ o (28 {n + 'I) (12 + k2)} {n + (9l2 + k2)} + 9f2V2 ] - )

where V( = {KH 0 / 4rcp}) denotes the Alfven velocity.

It is convenient to discuss equation (28) in non7dimensional from,

so that the important physical parameters of the problem may be

brought out clearly. Let us choose a dimensionless growth rate y and

a ~itne:µ~ionless wavenumber. x by meas ming n .and k in suita~le units.

We define:

x = kd -;;s' (29)

andy = nd nsV •

From eqns. (28), (29) and (30), we have

(163

(30:)

256y6 + 128y7 [4Ra + S (a2 + 2a + 16) ] + 64y6 {2R2 (a2 - 16) +

+ S2(a2-16) + 2RS(a3 + 2a2 + I6a+64)+ a+ 36 + 4 (SA-4Bx2)]

+ 32y5 [4R3 (a3 - 16a) + S3 (a4 - 256) + R2S(Sa4 + 18a3 + 64a2 +

224a - 256) + 2RS2 (2a4 + 9a3 + 48a2 + 112a) + 2R (a2 + 46a) +

2S (5a2 + 36a + 80) + 20A {4Ra + S (a - 3. 2)} - 16Bx2 (2Ra +

2Sa)J + 16y4 [R4 (a4-256) 2R3S(2a5 + 8a4 - 64a2-5I2a - 1000):,+

2R2S2 (6a5 + 19a4 + 64a3 - 512a + 256) + 2S3R (2a5 + 4a4 -.:

5l2a - 1000) + R2(Ila3 + 66a2 - 240a + 416) + s2 (lla3 + 4a2+

144.:i - 140) + l 2RS (22a3 + 108a2 + 96a + 460) + 10a + 369 +

2A {10R2 (6a2 - 32) + 2RS (1Sa2 -54a + 18) + 18} - I6Bx2

{R2 (6a2 - 32) + 8RSa2 + s2 (a2 - 16)] + 8y3 [R4S (a6 + 2a5 -

16a4-64a3-256a2 + 512a+4096) + 4R3 .Sf(2a6+ 5a5 - 32a3-512o2

- 744a) + R2 S3(a4 - 256) (6a2 + 16a - 32) + RS2 (36a4 - 11a3 +

+ 208a2 - l344a - 256) + 2R2 S (26a4 + 59a3 - 224a2 - 368a .,,......,

3072) + 5R3 (5a4 + JOa3 - 192a2 - 160a + 1692) + R (60a2 +

346a - 1952) + S (19a2 + 328a - 10008) + 2A {R2S (40a2 - 472a

+ 584) + R3(40:z3 - 64a) + 54Ra} - I6Bx2 {4R3 (a3 - 16a) + 2R2S

164)

(6a3 - 32a) + 4RS2 (a3 - 16a) + R (30a - 32) + S (lOa - 32)}]

+ 4y2 [R4 S2 (a2 - 16)2 (a3 + a2 + 16a - 16) + 4R3 S3 (a4 - 256)

(a2 - 16) (a+ 2) + R2S2 (3la5 - 12a4 - 512a3 + 160a2 - 1338a +

29696) + 2R3S(a2 - 16) (10a3 - 16a2 - 256) + 2RS (39a3 + 104a2+

1228a + 4730) + R2 (a2 - 16) (50a - 151) + 9 (a + 36) +

2A{10R4 (a2 __, 16).2 +2R3S (5a4 - 12a3 + 6a2 + 192a - 1376) +

18R2 (3Qa2 - 16)} - 16Bx2 {2R2S2 (a2 - 16) (3a2-16) +2RS(l5a2-

64a + 80) + 8R3S (a4 - 16a2) + R4 (a2 - 16)2 + R2 (30a.2..,.. 64a-

160) + 9}] + 2y [2R4S3 (a2 + 16)(a2 - 16)3 + R3S2 (a2 - 16)

(5g4 -64a3 + 160a2-1024a + 3840)+R2S (a2 - 16) (59a2 - 160a+

224) + 45R (a2 - 16) + 2A {R4S(a2 - 16)2 (lOa - 32) + 18R3(a3-

16a)} - 16Bx2 {2R3S2a (a2 - 16)2 + 2R2S (lla3 - 80a2 + 80a +

256) + 2SR4a (a2 - 16)2 + 2S2R2a (a2 - 16)2 + R3 (a2 - 16) (lOa

_,.. 32) + 18Ra}] - 16Bx2R2 (a2 - 16) {R2S2 (a2 - 16)2 + 2RS (5a2-

32a + 80) + 9} = 0, (31)

where

R = (rircsi2dV), .

S = (vrts/2dV),

B = (g~d2/rt2s2V2),

A = ( 4Q2d2/rt2s2V2),

(32)

(33)

(34)

(35)

(165

and a = 4x2 + 5. (36)

From the above equation we find that there are four parameters

required to specify y for any given x. These numbers R, S, B and A

respectively represent measures of resistivity of the medium, coeffici

ent of viscosity, bouyancy forces and coriolis forces in terms of magn

eti? field. The case of viscous, finitely conducting and non-rotating

configuration has been discussed in detail by Sharma [5], where the

nature of the boundaries chosen gives rise to an additiona.l monotoni

cally decreasing mode. Anoiher par~icular case of equation (31), when

the ,viscosity is absent, has been treated by Shar.ma [6]. The dispersion

relation (31) is an algebrica}equation of degree eight iny, hence it

will have eight roots. To obtain the explicitexpression for each value.

of y for general values of parameter is a task which is mathematically

too involved. If we consider equation (31) in its general form, we

observe that the stratification is stable or unstable thoroughly accord

ing as Bis negative or positive. However, if we consider the fluid to

be viscous but infinitely conducting (R = 0), the configuration is ren

dered unstable for the wavenumber range

• 112

1 [_41 I x < 2 <iB-5 j (37)

Thus the effect of resi5tivity is to cancel the stabilizing role of

magnetic field altogether and render the system umtable for the whole

wavenumber range.

We now examine the behaviour of growth rate with respect to

viscosity analytically To investigate this, we require to discuss the

nature of the positive root of yin equation (31) in detail. We observe

if B > 0, t.he absolute term in. equation (3 .1) is negative, therefore, the

166)

equation (31) has at least one real positive root. Hence, the equili

brium will be always unstable. The asymptotic behaviour of this

root for X __.,,. 0 and X __.,,. oo ate

y ___,. (9R2S2 + IORS + I) 3~ (9R2S2 + IORS + 1) (41RS + S) + AR2(9RS + 5) ( )

B y -i> --- • 2Sx2

(39)

Now we study the behaviour of y (growth rate) on varying the

value of S from equation (31). To find the role of S, we examine

the nature of dy/dS form (38), while keeping other nondimensional

numbers R, A and B fixed. To bring out the peculiar features of the

role of viscosity into focus, in the context of Rayleigh-Taylor insta

bility, we divide oi1r subsequent analysis of instability criterion into

two cases by different range of R and A.

Case 1. AR2 > l. In this case we find that with increasing S, the

value of y decreases for all wavenumbers. Thus, more the viscosity of

the medium, more it tends to stabilizes the configuration. This type of

behaviour can also be observed even in the absence of rotation.

Case II. AR2 < 1. In this case a peculiar tendency is exhibi ed by

s, when (a) s < s, (b) s > s~ where

S- _ _l_{-lOR -t- [IOOR2 - _!__ (82 - 9AR3 -- 18R2 82

( 82A2R4 + 5248AR2 )li2) 1112 J. (40)

So long as S < S, an increase in the value of S leads to the decr

ease in the value of y for all wavenumbers. Thus, we can say that

viscosity has a stabilizing influence on the configuration. When S

becomes smaller than S~ this behaviour is reversed for srnall values of

[167

x. Now an increase in the value of S leads to a increa'>e in the value

of y. This particular behaviour is however, marked only for small x.

After reasonably laqe values of x, once again the uniform pattern can

be observed, namely that Vvith incre~sing S. more stability is imparted

to the system. Thus, we conclude that viscosity has a destabilizing

influence for small wavenumbers of disturbanca but it has a stabilizing

influence for large wavenumbers.

Lastly, when B<O we find that all roots of y are either real and

negative or there are complex roots with negative real parts. The

system is therefore stable in each case. Hence, the potentially stable

configuration remains stable whether the effects of viscosity, resistivity

and rotation are included or not.

REFERENCES.

[I] S. Chandrasekhar, the character of the equilibrium of an incom

pressible heavy viscous fluid of variable density, Proc.

Carob. Phil. Soc. 51 (l955), 162-178.

(2] R. Hide, Waves in a heavy, viscous, incompressible, electrically

conducting fluid of variable density, Proc. Roy. Soc. London

Ser. A 233 (1955), 376-396.

[3 J Lord Rayleigh, Investigation of the character of the equilibriµm

of an incompressible heavy fluid of variable density, Scien

tific papers II, Cambridge, U. K., 1900, 200-207.

[4] B. M. Sharma and P. D. Ariel, The character of the equilibrium

of a heavy, viscous. incompressible, finitely conducting

rotating fluid in the presence of a vertical magnetic field,

168]

Indian J. Phys. 499 (1975), 100-116.

[5] B. M. Shar:ma, Waves in a heavy incompressible fluid Of finite

depth and of variable . density, Indian J. Pure Appl. Math.

5 (1977); 527-537.

[6] B. M. Sharma, Effects of electrically resistivity on the Ray}eigh:

Taylor problem with a vertical magnetic field,.{Jniv. Nae.

Tucuman Rev. Ser. A 30 (1982), to appear.

Jnanabha, Vol. 15, 1985

SOME INTEGRALS AND SERIES OF THE PRODUCT OE

TWO MULTIV ARIABLE H-FUNCTIONS

By

Y. N. Prasad and K. Nath.

Applied Mathematics Section,. Institute .of Techn'1fogy;.

Banaras Hindu University, Varanasi-221005, India

( Received: Decembu 5, 1983 ; Revised; July '5, 1984)

ABSTRACT

In the present pit.per, we evaluate three definite integrals. which are

then used to establish two Fourier sine series and one infinite series

for the multivariable fl-function due to H. M. Srivastava and R. Panda

[IO] .

1. INTR,ODUCTION

The multivariable H-function defined by Srivastava and Panda

[JO, p. 271, Eq. (4. l)] (see also .Prasad and Singh [7]and ·srivastava

and Panda [ 11 J) will be represented here in the manner already detailed

by Srivastava, Gupta and Goyal [9, p. 251, Eq. (C. I)]. The object of

th is paper is to evaluate three definite integrals '\\'.hich are then used to

establish a number of expansion formulas involving these multivariable

JI-functions.

2. Integral Formulas ,

In this section, we evaluate the following definite integrals

Jrr./2

0

' (sin x)u-1 (cos x)"-I e~(u + v)x H 0, 0: (1, N) ; ... ;

P, Q: [P, .Q' + l]; ... ;

170]

z Si W:» O"! (1, N<r>) [ . [p<r>,(2<r>+l] 1( nxe ) ; ... ;Z,(sinxeww)cr"I

{(e1l; Ev'; ... ; E1l<r>)} : {(Cv', Yv')} ; ... ;

{(/Q; Fg' , ... , Fo<r>)}: (Do', ao') {(Dg', 8'g')} ; ... ;

{(C<r>p<rl • Y1'1p<r> )} J

(Do<r> ' ao<r>) ' {(D<r> Q<•> ' a<•> Q<r>

0 0 · (V' W') · · (V<•> W1' 1) r · ' " ' , ... , ' w h H lx1 (sinxe 0

) 1

; ... ; p, q: [X', Y'] ; ... ; [ _.f<r>, y<r>J

Xr (sinx ew.c)h' I {(aj); rJ.j)' , ... , aj)<'>)} : {(A'X' • "YJ'x• )} ; ... ;

k(hq; {3/ ,. •. , {3q<•>)}: {(B'y 1 , ~· Y' )} ; ... ;

{(A<tJx<r> ' "fJ<r>x<r> )} 1 < > J dx. {(Blr)y(r) • & r y<•l)}

ewun/2 r(~)_ ~ tfo(pV1 , ... , Pvr) = I '> (t) V --0 0 ... oo Vt ,... r -

r w ( ~ O"~ p ' )11'/2

e i=l i. r fl { 6; (pVt )

i=l

(-l)v' z1

Pvt} H 0, 1 : (V', W') ; ... ; (v<r>, W<rl)

V; ! p + 1, q + l; [X', Y'J ; ... ; [X<rl, y<rl]

[x ewh1 n 12 wh, n/2 (l - u. - .~ 0-1 Pv ; hi, ... , hr)

1 . , ... ,x, e i=l • {(hqj {3'q , ••• , 13<rlq)},

{( ' . (r) )} • {(A' . ' ) } • • ap; cit p ,. •• ,cit p • X' , "IJ X' , ... , r

(1-u-v- ~ cup . ; h1 , •.. ,hr): (B' Y', ~' Y') ; ... ; i=I v,

{ ( A<•>x<r> , "IJ<r>X<•> )} J-. , {(Blrly(r) '~(r)y<rl )}

r provided that w = .Y-1, hi. h2 ,. .. ,hr > 0, Re (u + ~ (ai Pv

i=l i

+ lu cit;)) > 0, Re(v) > 0, I arg Xi I < t U;n, U; > 0,

I arg z, I < t V;rc, Vi > 0 (i = 1 , .. ., r),

where a.;, U; and Vi are as follows :

[171

(2.1)

a;= min Re (B/il I ~/il),j = l ,. .. , vrn, i = l ... ,., r, (2.2)

rnd

p q vw y1;J

U, = - ~ a.;<O - ~ ~;(il + ~ ~;(i) - ~ ~j(i) + j=l j=l j=l j=Vli>+l

xw ww ~

j=l "lj;<;1 _. ~·. ri;w > 0, i = l , ... , r, (2.3)

p Vi = - ~ E; -

j=l

. j=WW+l

Q QW ~ F; + 80 ~ 8/i> +

j=l j=l

pw

NW ~· y;<il

j=l

~ y1 > 0, i = I , ... , r. (2.4) j=N(i)+l

Jn/2

0 (sin x)u-1 (cos x)~-1 ew(u+v)x H 0,.0 : (I + N') ;; .. ;

P, Q : [P'' Q' + 1) • • ' ••• >

172]

(l, N<rl) [ [PM, Q<rl + l] Zi(tan x)cr

1 , ... ,Zr {tan x)O'r

I {(ep ; E'r , ... , £<r>pM ) } :

{(/ Q; p· Q' , .. ., F<rl Q{r) ) }:

{( C'p, , y'P' )} ; ... ; {(CirlpCr! • Y'''pcri )} J {( D'o' a'o ), {(D'Q, a• Q' )} ; ... ; (Do''i, ao<rl) {(D<rl Q<rl ,a<rl Q<rl )}

0,0:{V', W'); ... ;(.v<rl, w•rl) r h h I H . I X 1 (tan x) 1, Xr (tan x) r

p. q: [X', Y'] ; ... ; [X<rl, ycrl] L

{( . ' (rl)} • {(A' '. ' )} • • {(Air) (•) )} 'l Gp, ex p, .. .,ap,.. . X' ' YJ X' , ... , x<r) ' "IJ x<1l (b . P.' P. (rl) . {( ~' y 1 )}· • • {(BCrl : ~Cr) )} Jdx. q, ,..q, ... ,pa · "Y'' sy1 , ... , ycr) '" ycrl

ewu~2 = I'(u + v) ao' ... ao<rl ~

V1 , ... , Vr ~ 0 ef>(p ' .. ' p )

Vi Vr

r cv( ~ .cri Pv. )rr./2

e · i=l · ' r ( 1 )w p . 11 {~; (p . ) ---=- Zi Vi }

i=] Vi Vi !

1., .1 : (V', W') ; ... ;.(V<ri, w<rl) [ whirc/2 wh,rr./2 H . . x1 e , , ... , Xr e

+. 1 +'l . [X' Y•J . . [X<rl y<rl] . p 'q .. ,, , .. ., ' ..

,. (I - u - ~ cr,p . ; hi; ... ; hr), {(ap; a'p , .. ., a<rlP)}: {(A'X'' ri'x1)};

i=l v, . .

r {(ba;· ~ 1 q , ... ; ~<rlq)}, (v- .~ l O'i Pv;; h! , ... ~hr: {(B' Y', ~· Y')} ;

l=

ti7~

•.. ; {(A<r>X<r> ' r;<'>x<r> )} l • {(Bfr) l:(r) )} j ... , y<r> • <., y<r> •

w = .Y-1, (2.5)

r provided that hi. h2 , ... , hr > 0, Re (u + :Z (crt Pv. + hJa.;)) >0,

i=l •

r Re (v - :Z (cri Pv + hJ ~;)) > 0, I arg x; I <: ! U,'lt, U; > 0,

i= 1 j

I arg Z;, I 0 (i = 1 ,. .. , r), where r:t..i, Ui and V. are

given by the Equation (2.2), (2.3) and (2.4) respectively and

~i =max Re ((A/il - 1)/'1J;<t>), j = 1 , ... , ww, i = 1 , ... , r

(2.6)

xP e-x V">(x) H Ioo 0, 0: (1, N') ; ... ;(I, N<rl)

O 1o P, Q : [P', Q' + l] ; ... ; [p<r>, Q<r> + 1)

r cr1 Cir I {(ep; E'p ,. .. , E(r)p )} : {(C'r• y'p-)}; ... ;

LZ1 (x) , ... , Zdx) I · . -· {(/Q; F'Q , ... , Fd'>)}:(D'o,3'o),{(D'Q',3'Q')}; ... ;

{(CCT)p<r> ''{(r)p<r> )} I (Do<r>, 3o<'>) , {(Dir> QC•> , 3<rl QCrJ )} J

H 0, 0: (V', W') ; ... ; (V<r>, wcr>) r p, q: [X', Y'] ; ... ; [Xfr>, y<rl] L C1 (x)f''l ; ... ;Cr (x)"'rl

I

{( • ' C1'l)} • {(A' · ' )}• · {(AC•> .,,er> )}

I. Gp,.r:J.p. , ... __ ·_°'P.· • X'•1lx• ,._ .. ,. x<r>•···x<r.) 1

{(b . A ' A (rl)\ • {(B' . i:' )} • • {(B<r> l:(r) )} J dx. ~· 1"'1 .. ""•l"'q J • Y' • 'o y1 , ... , y<r> • 'o y<rl .

~o/~1

00

krno' ... ao<rl- . ~' o· <fo(pv , ... , p ) Vt ,. .. Vr = 1 Vr

1

r Vi p TI {0; (p ) (- I) Z 1 Vi }

i= 1 V; I'; !

1, 1 : (V', W') ; ... ; (V(r>, W!r>) [ H Ci

p + 2, q + I : [X', Y'] ; ... ; [X<ri , y<r>] , ... ,, Cr I

I'

(cr - p ·- ,·.~ 'ci'ip . ; µ1 ; ... ; µr), {(a~; a'p ,.:., a<r)p )} ' i=l v •

. .. r

(- cr - p ~ <1i p . + k; µi ,. . ., p.r), i= 1 Vi

r ( - p - ~ <1i p : µi ,. . ., µr) :

i=I v.

{(bq; ~·q ,. . ., ~<r>q )} : {(B'y,, ~'y )} :

{(A'x•' 'IJ'X' )} ; ... ; {(ACrlx<•> 'IJ<•>x<•> )} ] · • f(B<r) !:(r) )}

, ... , ·l y(r) • <., y<r> ,

r provided that Re (p + I + ~ (tT• p . + µtext))> 0,

i= I v,

(2.7)

I are c, I < -! Utrr. , U; > 0, I arg Zi I < t Virt , Vi> 0 (i = I,. . .,r),

where a;, (Ji and Vi are given by the Equations (2.2), (2.3) and (2.4)

respectively.

Proof. To prove (2.1), we write the series expansion of

H<l> [Z1 (sin x ewx )a1 ,. .. , Zr (sin x ewx )<>r with the help of the result

(175

given by Sax~na [8)and Mukherjee and>prasad [6,p.6], as follows,;

srn [Z1 (sin x ewx )a1 , ... , Z,. (sin x ewx )a']

1 ~ efi(p , ... , Pv,) -. - -0. V1 V1,·· 0 ,Vr - ,

00

r ( l)vi p a·p Il {0;(p )----• Z; V; (sin X ewx ) i Vi} p _Do<il + Vi

i = I Vi v, ! ' Vi - ---.-..

w=if-1, (2.8)

where

and

N<il n ro - c1 + y/i> s;)

j=I 0;(s;) = -Q(i) pw n r(I - D;w + 8/i's;)

j=l n r(C/i> - y/i>s,)

j=N(i>+I

i = 1 , ... , r, (2.9)

[

p r Q rp(s1 •..• , Sr) ~ rr r(e; - ~ E/il s;) n r (l - f; +

j=l i=l j=l

r J-1 . ~ F;W Si)

l=l ' (2.10)

where {a;)> 0, I arg z, J < 1/2 V; n, V; > 0 (i = I., ... , r),

where V; is given by Equation (2.4); take the contour integral form for

H'2'[x1(sin x e>;~l1 , ••. , Xr (sin x e"'"')hr] = -(2

1 ) [, ···I· rfo __ i(S1) . ~·)4 L '

.1761

.. ;efl;. (s.:.)'Y (si..:.,s.) x/:lc (sin•x ewx )hisi ... x/• (sin x lJJX )hrsr

ds1 .. dsr, w = if-1, 'c ' ', •

(2.11)

where

vw w<i) . n r (B;(i) - r;1w si) n r (I - A/i> + 1J;w s1) j=l j=l

eflt(s;) = yw xw· n r(l-B1(i) + ~;Ul Si) n r(A/il_'lj;(i) Si)

j=V<il+I j=WW+I .

• i = 1 , ... , r,

'Y(s1 ,. . ., Sr) = r rt l j=l

r r(l - b; + ~ ~/il St)

i=l

p r . . l~ Il r(a; - ~ «/'> S;) J

j=l i= 1 ,

(2.12)

(2.13)

in the integrand of left hand side of (2.1) and change the order of inte

gration, which is justifiable under the given conditions. The left-hand

side of (2. ,1) reduces to .

/ 1 ·- ~ ..!.( <lo'- .. ao<r> 'I' Pv , .. ., Pv) Vi.···•Vr=O 1 r

00 r Vi p Il { 0.(p . ) (.:::.!l_ Z; Vi }

i=l V1 Vi !

___ l -J .... r <foi(s1) ···.rpr(.sr) 'F'(s1 , ... ,Sr) L

1 }t,. . . :

S1 Sr X1 •·· Xr

r u + ~ (a; p . + h1s;) - 1

{ fn/2 . i=I v. Jo (sin ·x) (cos x)~-1

[177

r w(u + v + ~ ( cr; p . + h;s; ))

. 1 Vi l 1

= dx ~ ds1 ... ds,. (2.14) e

Now, evaluating the inner integral in (2.14) with the help of the result

[4, p. 80, Eq. (7))

J7t/2

(sin x)•d (cos x)rl ew(u + v)x d _ 0 X-

r(u) r( V) WU 1t/2 ~'----''--" e ' r(u+v) -

(2.15) Re(u) > 0, Re (v) > 0, w = v'"-1,

we arrive at the right hand side of (2.1) stated as above.

The proof of (2.5) is similar to the proof of (2.1 ).

To prove (2.7),we write the series expansion of fl!V[ Z1(x)cr1 ,. . .,Z,(x) crrJ

from the Equation (2.8) and contour integral form of fi!2>[C1(x)'"'\

... , C.(x)i-'') from the Equation (2.11) in the left-hand side of (2.7)

and change the order of integration, which is justifiable under given

conditions. The left-hand side of (2.7) reduces to

~o' ... 8o<r> 2: _ cp(pv1 , .. ., Pv,) Vi, ... ,vr-0

1 00 r Vi p

l1 { 0.(p ) (- l) Zi Vi } i=} V1 Vi !

(2~w)' JL1 ... IL. <h(s1) ... cpr(sr) lf(s1 , ... ,Sr) c/1 ... c/'

;

p + 2: (cr; p . + µ;Si)

{ f 00 i= 1 v, }

0 x e-i!Jv~>,. (x) dx ds1 , .•. , ds,.

(2.16)

178]

Now, on evaluating the inner integral of (2.16) with the help ofthe

known result [3, p. 292, Eq. ( 1 )]

f00

xl!-1 e-re Li°'> (x) dx = r(ot - ~ + k + l) r(~) , Re(~) > 0, J0 k ! r{a - ~ + 1)

we arrive at the right-hand side of (2.7).

3. INFINlTE SERIES

In this section, we establish aa infinite series for the product of

two multivariable H-functions. The series is

oo x2a1 - 2- t Wt 0, 0: (1, N') ; ... ; (1, N< 1 1) ~. .-·-·-· H

t=O 1 ! P, Q : [P', Q' + lJ; ... ;[P<•>, Q<•> + 1]

~ 2E ,, 2E (i-) 1·.{(ep .; E'p ,. .. , Emp.·. )'}: {(C'r· y/p. )}; ... ;

Z1(x) 1 ,. . .,z, (x) 1 . ; {( " · F' v <'')}·(D' "'' )'{(D' '.,.., ')}· , ;Q, Q,. . .,rQ . o,o o Q ,o Q , ...

{(c(r) . "'( r) )} _ p<r> '1 p<r> ·

(Do'iJ, 8o<rl) , {(D<r> Q<T> , ~(r) Qfr) )} J

0 . (V' rrJ•) . . (V<rl w<rl) f. I 'n . ' rv , .. ., ' . -2a.1' -2a.1<rJ I

H C1 (x) ; ... ;Cr (x) p, q: [X'' Y'] ; .. ; [ x<r>_, y<r11 L

(a1·- t, 0ti' ,. .. , a1!1>){a;, a;', .. ., a;< 1 ')2;p;

{(bq; ~q' , , , ~,1 <rl)} ;

{(A •.· . ' )}• . {(Alr) (r) )} l X' ' 'l X' "." X< 1 > ' 'IJ X< 1 >

{(B'y•1' ~· Y' )} ; ... ; {(B<r>y<rl '~(r)y(r) )} J

[179

= (x2 _ ~Vx)acI 1 00

~ 80' ••· 8o<r> . ¢(pv , ... , Pv ) Vt,.··· Vr ·=0 1. . .r .

. ~ {!!; (p . ) (- I)Vi z/v; (x)2£1Ci) } l=} v. Vi !

0, n: (V', W') ; ... ; (v<rl, w<r>)[ . . I ( 2 . -«1 H ~x-~) ,

p q: [X', Y'J ; ... ; [X<r>' Y•'>J

••• , r X - X · ' C ( 2 W )-ix1<r> { {(ap; a'p , .. ., a<r>P )} : {(A'x,' '1J'x, )}

· {(bq; ~'q ,. .. , ~<r>a )} : {(B'Y', ~'y )} :

• {( &Cr) .,,Cr> )} J •• ., r.i. x<r> ' ., x<1> . • • (r) (r) , .. ., {(B ycrl ' ~ y<r> )} ,

{3:1)

provided that r w/x I < L I arg C; I 0, I arg Z1 I < ! Vm, Vi > 0, where U; and· Vi are given by the Equations {2.3)

and (2.4) respectively, and (a,; a;' ,. .. ,, ClG;< 1 >)2,p stands for the set of para

meters {a2; a 2 ' ,. .. , ClG 2<'>) ,. •• , (ap; 1Xp 1; , ... , ctp''>).

Proof: To prove the result (3.1), we write the series, expansion of

2E' 2E <r> H<l> [Z1 (x) 1 ,. . ., Z, (x) 1 J with the help of equation {2.8) and

· -2a1' -2a1 Crl wnte H<~> [C1 (x) , .. ., C, (x) ] into its contour form, cha-

nge the order of integration and summation and use the formula

1F:.o (a; - ;x) = {I - xra, and-finally interpret the result in the light

of multivariable H-function.

PARTICULAR CASES

( 1) On taking r = 1, P = Q = 0 = P' = Q' = Do', 80 ' = 1 = E1 '

180)

2£' 2£ (r) z1

.....,,. 0 in 1f<11 [Z1 {x) ~i , ... , z,. (x) 1 ] in the Equation (3.1) .

we arrive at the result recently studied by Maurya [5, p. 255].

(2) On taking r = 2 in the particular case { l ), we arrive at the result

given by Chaurasia [2].

Acknowledgement

Our thanks are due to Professor H. M. Srivastava, University of

Victoria, B. C., Canada, for his valuable suggestion in preparation of

this paper.

REFERENCES

[I] B. L. J. Braaksma, Asymptotic expansions and Analytic contin

uations for a class of Barnes-Integrals, Compositio Math.

15 (1962) 239-341.

[2] V. B. L. Chaurasia, On some infinite series of H-function of two

variables, Vijnana Parishad Anusandhan Patrika, 20 (l 977),

91 95.

[3J A Erdelyi, et al, Tables of Integral Transforms, Vol. II, McGraw

Hili, New York, 1954.

[4] Erdelyi, et al, Higher Transcendental Functions, Vol I. McGraw

Hill, New York, 1953.

[SJ R. P. Maurya, A Study on a Generalized Multi-dimensional

Integral Transform, Ph. D. The'>is, Banaras Hindu Univer

sity, 1979.

[6] S. N. Mukherjee, and Y. N. Prasad, Some infinite integrals invo

lving the product of fl-functions. Math. Education 5 {1971),

5-12.

[181

/'i J y. N. Prasad and A K. Singh, Application of H-function of seve

ral complex variables in production of heat in a cylinder,

Pure Appl. Math. Sci. 6 (1977), 57-64.

[8] R. K. Saxena, On the H-function of n variables, Kyungyook

Math. J. 17 (1977), 221-226.

[9] H. M. Srivastava, K. C Gupta, and S. P. Goyal, The JI-functions

of One and Two variables with Applications, South Asian

Publishers, New Delhi and Madras, 1982.

[10] H M Srivastava and R. Panda, Some bilateral generating

functions for a class of generalized hypergeometric polyno

mials, J. Reine Angew. Math. 283/284 (1976), 265-274.

[11] H M. Srivastava and R. Panda, Certain multidimensional inte

gral transformations/, Nederl. Akad. Wetensch. Proc. Ser.

A. 81(1978),118-131.


FINITE INTEGRALS INVOLVING THE PRODUCTS OE

THE H-FUNCTIONS OF TWO AND MORE VARIABLES

By

R. M. JAIN

Department of Mathematics, University, of Rajasthan,

Jaipur-302004, India

t Rece il'ed : December I 3, 1984 )

ABSTRACT

In this paper we evaluate two finite integrals associated with the

product of the H-function of two variables and Srivastava and Panda's

multi variable H-function. These integrals are quite general in character

and a number of interesting (known or ne,v) integrals can be deduced

as particular c1ses. -Some special cases are also discussed briefly.

1. INTRODUCTION

The H-fur.ction of two variables occurring in this paper is due to

Mittal and Gupta [6]. The details of this function can be found in a

recent book by Srivastava, Gupta and Goyal [11].

The following expansion formula ([4], p. 17, Eq. (2.1)) for a

special H-function of two variabl~s will be required in the sequel :

0, n1 ' : 1, n2 ' : 1, n3 '

(I I) H* [ax bx] = H , , ' + l · p3' qg' + 1 • , ' P1,' q1 : P2 ' q2 ' '

[

ax I (u;; µ;, U;)l,Pt' : (g;, G;)l,p2' ; (k;, K;)l,pa' J bx (v; ; '1Ji, V;)l q ,: (O, 1), (h;, H;)l q , ; (O,l), (I;, L;)l q ,

' 1 ' 2 ' 3

184]

00 I; h (-ax)M

M .. -.. -.-M=O M! ..

where

'M (1.2) hM =

1

:2: cj;'(M- N, N) 01 ' (M- N) 82' (N)(b/a )1'(1jf) N=O

n1' (l.3) g,' (~, ") = :2: r(l - u; + µ; F, + U; ,\)

j=I

[

P1 1 qi' J-1

II r(u; - µ;F, - U;'A) fl r(l - VJ + ·.i;F, + V;,\)J j=n1 '+l )=I

n2' tl4) 01' (0 ·= fI f(l -g; t G;F,)

j=l

r,.. P2' lI r(g; - G1~)

LJ = n2' + 1

q2' -1-1

n rl1 - h; + H1F,) , J=l J

and with 02 '(,\)defined analogously to 01 {F,) in terms of the parameter

sets (k;, K1)1 , , {It, L1)1 , . ,p3 ,q3

The H-function of several complex variables (or the multi variable

H-function) v,as introduced and ~tudied syste.matically in a series of

papers by Srivastava al).d Panda (see, e.g., [12] and [13]; see also

[11] ). We shall employ the following contracted notation of Srivastava

and Panda ([13], p. 130, Eq. (1.3)):

-. 0, n: m1; nr, ; ... ; mr, nr [~1 1 (1.5) Hfz1, .. .,z,]=H :

p, q : Pl, ql ; ,. · ; Pr, q, Zr

{185

( • / .<r>) . (c' y·') . . (c·irl .,.frl) I a; , rt.; , •.. , a, i,p . ; , 1 1 , · ·, 1 , ! ' 1 J ,pl ,pr

(b . . A ' r.>. (r)) • (d·' '>.) · · (d·(rl <:-.(r)) 1 , t'i , •.. , t'J Lq • 1' o, l , ... , 1 ' 0 1 1 ,q1 ,q,.

to denote the fl-function of r complex variables z1 , ... , Zr. [See

Srivastava, Gupta and Goyal ([I IJ, p. 251, Eq. (4.1)) for details of

this function. J

With a view to facilitating the derivation of our main integral

(2. l) in the next section, we give here an elementary integral contained

in the following

Lemma. If

(i) min \ 1 < j < r r;, p;, Re (r1.), Re(~)} > 0,

r r Re (a) + k cr; ~; > 0, Re(()) + ~ p; ~i > O,

j=I j=l

where

( \.6) ~; = 1

..,.... m.i1:- [Re (d/!, / 8/kl) ] ""=:} """m"

V kc: {l , ... , r},

and

(ii) U1c > 0, j arg Zk I > t u,, 'It,

where p q llk

(l.7) U1, = - k a/k> - k j = n + l j=l

();<k) + ~ y;(k)

j=l

P1< mk qk .E y/kl + k ?)/kl - ~ 8/M

j = n1, + I j=l j = m1c + 1 V kc: {l , ... , r},

186]

then

1rc/2

(1.8) 0

exp (i (a. + ~) 0} sinrl 0 coslH 0

where

( 1.9)

H [z1 exp { i( cr1 + p1) 0 } sin cri cosP 1 0 , ... ,

Zr exp { i(crr + pr) 0} sincr' 0 cosP' ll] d0

= exp (irca./2) H ~l,h [z1 exp (ire cr1/2) ,. . ., Zr exp (ire cr,/2) ]

H {l) <><'fl

0, n + 2 : ml> ni ; ... ; mr, nr [z1 ,. . ., Zr]= H

p + 2, q + 1 : P1, q1 ; ... ;pr, q,

l?j (1 - a ; CJ1 , ••. , CJr), {l - ~;Pt,. . ., pr),

Lzr (1 ~ r:J. - ~ ; CJ1 + P1 ,. .. , CJr + P•),

' (d) (a; ; rJ.; ,. ·., «1 l.p • ( ' ') ( (r) (rl) • C; 'Yi I , ... , C; 'y; 1 ·1 1Pl ,pr

(b;; ~/ ,. ... ~/·»1,q . (d I " ') . . (d (r) " (r)) J • j ' Oj 1 . ,. . ., j ' Oj I ~-' ,ql ,qr

Proof. To establish (1.8), we first write the Mellin-Barnes contour

integral ( [ 11 ], p. 251, Eq. ( 4.1 )) and change the order of integrations

tberein We then apply MacRobert's result( [5], p. 450, Eq (4) ):

Jrc/2

(1.10) exp {i (a + ~) 0} sin..-1 0 cosl3-1 0 d0 0 ..

= exp (irr.a./2) B (a, ~),

{Re(a.) > 0, Re(~) > O},

in order to evaluate the 0-integral, and interpret the resulting contour

L187

integral as the multivariable H-function; the integral (1.8) follows at

once.

2. THE MAIN INTEGRALS

We shall establish the folloNing two general and finite integrals

; r7t12 (2.1) Jo exp { i(a + ~) 0} sin"-1 0cos1H e

JI* (a exp { i(cr + p) 0} Sino-() COSP 0,

b exp { i(cr + p) 0} sino- 0 cosP 0]

H fz1 exp { i(cr1 + p1) 0} sincr1, 0 cosP1 0 , ... ,

Zr exp { i(crr-!- Pr)0} sincrr 0 .cosPr 0] d0

~ hM (-a)M exp {i 7t (a+ crM)/2} M=O· M!

(I) H f z1 exp (i 7t ril/2) ,. , . , Zr exp (i 7t crr/2) ] ix+crM, ~+pM

The integral (2.1) is valid if the following (sufficient) conditions

are satisfied : ~

(i) The sets of concitions (i) and (ii) mentioned with the Lemma

hold true

(ii) p > 0, cr > 0.: U G Q, and V ;G O~where

qi' (2.2) u = 1 + ~ "f)j +

j=l

,q2'. ~ H;

j=l

P11

~ µ; -j=l

P21

~ G1 j=l

188]

q1' q3' Pt I (2.3) V = 1 + :l: V; + I; Li - :l:

Pa' Ui- :l: Ki

j=l j=l j=l j=1

where each of the equalities holds when the variables are suitably con

strained.

(iii) The series occurring on the right-hand side of (2.1) is absolutely

convergent.

(2.4) Jt x"-1 (t -~ x)ll-1 (1 + sxr""-11 0 .

l!* [a xct (t - x)P bx" (t - x)PJ (I + Bx)"+P ' (1 + Bx)"+P

[

I X a. t-x p1 · x Gr t-x Pr :

H z1 ( --) • (---) , ... ,Zr\--) (--) idx I+Bx I+Bx 1-tBx I+Bx J

t"'+B-1 00 (-a)M t"+P · . ~ hM -- (---)M

(I+Bt)"' M=O M! (I+Bt)"

r 1a1+P1 (l LZ1 ·---- (J Ha.+aM,~+pM (l+Bt) 1

, ... ,Zr tar+pr r J

The integral formula (2.4) is valid under the following condit

ions :

(i) Bt > - l

(ii) conditions (i) and (ii) given with the Lemma hold true

{iii) The series occurring on the right-hand side of (2.4) is absolutely

convergent.

[189

In(2. 1) and (2.4), +1*-fonction, rillJ -fonction aridhMare defined a,"

by (1.1), (1.9) and (1.2), respectively.

Derivation of the integrals. To establish the integral (2.1 ), we use

the series expansion Cl.I) for H*-fi.frction occurring in , the integrand,

change the order of integration and summation (which is justified

under the conditions given with (2.1) and evaluate the resulting

0-integral with the help of (1.8). we easily arrive at the right hand side

of(2.l).

The proof of the integral formula (2.4) is similar to that of the

integraF(2;·1) with·tne only differencdhat 'here·we use a •known inte

gral due to Agal and Kou! ( f !], p. 14, Eq (2.1)) imtead of (I.8).

3 Special Cases The multivariable H-function is an extension of the

G-function of several complex variables, and includes Fox:s H;fuhction

of or.e and two variables, Meijer's G-function of one and two varia-

bles, the generalized Latricella function of Srivastava and Daoust

[10], Appell ftnctions, the Whittaker functions, etc. (cf.[11]).

Therefore, our integrals can be suitably applied to evaluate the corr-

esponding integrals involving these functions simply by specializing

the parameters of the multivariable H-functions.

On the other hand, on specialization of the ; parameters of the

'H'i'[ax, lt.x'] ~ilitably,·several 1rifegrals involving the various products

of elementary functions of one or two variables and the multivariable

H-function can be easily obtained. Thus, fof example, if we

n1 ' ~ p 1 ', n2 ' = P2', n3' = p3' in (2 1) an:d.1;se"the kriowrrresult

p. 88, Eq. (6. 4. 2)), \\e get

,{

190]

rrc/2 (3.1) Jo exp { i(a + ~) 0} sinc.-1 0 coslH 0,

S[a exp {i(cr + p)0} sin"e cosP 0,b exp {i(cr+p)8} sin"" 0 cosP 6]

H[z1 exp {i(cr1 + Pt) 0} sincr1 0 cosP1 0 , ... ,

Zr exp {i(crr + pr) 0} sincr' 0 ccosPr 0 Jd8

oo M . = ~ h 'M __!!_. exp {i n(a + crM)/2}

M=O Ml

f]c1) r c· a-j-crM, ~+pM z1 exp l re cr1/2) , ... ,Zr exp (ire cr,/2)]

\\-here

(3.2) S[x, yJ = S Pi, : P2' ; p3' [(u1 ; µ1 ' Uj)1,Pi' :

q I • I I 1 · q2 ; q3 _( Vj ; 'fjj , Vi) ,

. . 1,ql :

(g1' Gj)l p I (k/, Kj)I p3'· ·1 ' 2 ' '' x y J

(111 , H1)1 , (/;, L1) 1• ~ '· 'q2 . '13 '

stands for the (Srivastava and Daoust) generalized Kampe de Feriet

function ([9], p. 199, Eq. (2.l)),

M . CU) h~' = 2::. <fi"(M - N, N) 01"(iVI - N) 02"(N) (b/a)N ( MN ),

N=O .

<P" (~, ,\) = I <P' (~, ,\) j ni' = P1 1 '

Ei" (~) = J e~' (0 /n2' = P2 1

and

[191

02" (,\) = I 02' (,\) 1n3

' = Ps' ·

The condition of validity for the integral are the same as menti

oned with the main integral (2.1).

Again, setting n1' = p 1 ' =qi' = 0 in (2.1) and .using the known

result ([10], p. 90, Eq. (6. 4. 15)), we arrive at the following interesting

integral :

f it/2 (3 .4) j

0 exp { i(a + (3) 0} sinrl 0 cosB-1 e

l,n2' r . 0 1(g1,G1)1 , J H

1

, I a exp {i(a + p)0} sin.,. cosP l ,p2 P2' q2 + 1 L (O,l),(h1,H1)1,q2'

1, ns' [ l (k1, K1)1 , J H , . , b.exp {i(a + p)0} sin.,.cosP 0 ,p3.

p3 , qs + 1 (0,1), (!1, Lj)l,qs'

H {z1 exp { i{ci-1 + P1) 0} sina1 0 cosP1 0 , ... ,

Zr exp { i(crr + p,) 0} sina,. 0 cosP' 0J d 0

oo (- )M :E hM" _a_, exp {ire (a+ aM)/2}

M=O M.

IJ(l) . a+a.M, f3+pM [z1 exp (z rc cr1/2) , ... ,Zr exp (ire a,/2)]

where

M (3.5) hM'1 = :E 01' (M - N) 02 (N) (b/a)N (M/N)

N=O · · · · · · ··

1921

Similar type of integrals can be obtairted from the integral (2.4)

also.

'Lastly, 'if \Ve put ins' :..:.... Ps' 'd:::'(j3 1 '==;b 'in '(3'.4) arid let b--?>- 0

therein, we arrive at the known restdt due to Gat'g ([3], p. 06, Eq.

(2.1)). The results established earlier by Rathie ([8], p. 237, Eq. (2.3)), 'vliMsffat and'Goyal'['14J, p. 13, Eq. (Ll )), and others, can also be ded-. . 'uted as'sf;ecialcases ~((2.1). Also, 'the known integrals due to Garg

([2], p. 23, Eq. (2 1)), Agal and Koul {[l], p. 14, Eq. (2.1)), Prasad

and Singh ((7], p. 126, Eq. (2.1), and Srivastava and Paooa ([13], p.

131, Eq. (2.2)), follow easily'as ·spedal 'cases of our second integral

(2 4).

:A:<?knO'wledgemettts

The author is grateful to Dr. S. P. Goyal for his generous help

and guidance in the preparation of this paper. He is also thankful to

)::>rof. ~.'C .. Gupta''a1~d-Ptbf.'H.·M. Srivast'a'.va (Canada) fodheir very

valuable s'tiggcsti6ns and encouragement.

;R:EFERENCES

(IJ S. N. Agal and C. L. Koul, Integration of certain products

associated \Vith the H-f unction of several variables, rnanabha

12 (1982), 13-19.

[2] R. S. Garg, Some integrals involving the inultivari'able If-function,

Jfianabha 11 (1981), 21-29.

[3] R. S. Garg, Some general integral relatrons'for the multivariable

If-function, Jfianabha 11 (1981 ), 133-145.

[4] S. L. Kalla, S. p. Goyal and R. K. Agrawal,\On multiple integ

ral b.'artsf6rm'afioiis, Wlath. fvotae'2s (197'7), i s-i;7. ·

[193

[5] T. M. MacRobert, Beta function formulae and integrals involving £-functions, 1'vfath Ann, 142 (1961), 450-452.

[ 6] P. K. Mittal and K. C. Gupta, An integral involving generalized function of two variables, Proc. Indian Acad. Sci. Sect. A 75 (1972). 117-123.

[7] Y. N. Prasad and S. N. s:ngh, Some expamion formulae for the H-function of two variables involving Jacobi polynomials Bull Math. Soc Sci. Math. R. S. Roumanie (N. S) 21 (69) (1977), 123-132.

[8] A K. Rathie, An integral involving H-function. II, Vijnana Parishad Anusandan Patrika 22 (1979), 235-238.

[9] H. M. Srivastava and M. C. Daoust, On Eulerian integrals associated with Kampe de Feriet's function Pub!. Inst. Math. (Beograd) (N . ..'> .) 9 (23) (1969), 199-202.

[IO] H. M. Srivastava and M. C. Daoust, Certain generalized Neumann expansions associated with the Kampe de Feriet function, Nederl. Akad. Wetensch. Proc. Ser, A 72 = Indag. Math. 31 (1969), 449-457.

[11] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Tv. o Variables irith Applications, South Asian Publishers, New Delhi and Madras, 1982.

[ 12] H. M. Srivastava and R. Panda, Some bilateral generating functions for a class of generalized hypergeometric polynomials J. Reine Angew. Math. 283/284 (1976), 265-274.

[13] H. M. Srivastava and R. Panda, Expansion theorems fo1 the JI-function of several complex variables, J. Reine Angew. Math. 288 (1976), 129-145.

114] S. K. Vasishta and S P. Goyal, Sorlie finite integral involving the H-function of two variables, Univ. Stud. Math. (Jaipur) 3 (1975-76), 13-22.

jfianabha, Vol. 15, 1985

CERATIN APPLICATIONS OF JACKSON;S SUMMATION

FORMULA

By

V. K. JAIN

Department of Mathematics, Bareilly College

Bareilly-243001, U. P., India

( Received: January 17, 1983; Revised: March 12, 1984)

ABSTRACT

In this note we show that the q-Watson and q-Whipple summations

due to Andrews are the particular cases of the well known Jackson's

formula for summing 8<!>7. We also obtain the identities of Rogers

Raman ujan type related to' the modulus 23.

1. INTRODUCTION

lf we let

I q I < I, [a; q]n =(I-a) (l-aq) ..... (1 - aqn-1), [a; q]o = 1,

(1.1)

00

[a; q] = IT (1 - aqi) 00 • 0 ]=

(1.2)

then the basic hypergeometric series may be defined by

P+l<l>P+r = l:: [a1, _qJ,. ... [aP+l; q],. (-)"rxn

[

a1 , ... , aP+l ; q;x] oo •

b1 ,. .. , bv+r n=O [q;q],. [b1; q],. .. [b11+r;qr,;--.

.qrn<n-V/2 (1.3)

where the series converges for all positive integral values of rand for

196]

all x, except that, when r = 0, it converges only for I x I <1 1. Further,

we sha1J 1simply write

ra1 , .. 'ar; q] IT ! .. h1 , ... , hs

for ~ (a - a1 qD .... (1 - a, qi) j = O (I - b1 q1) .... (I - bs q ;) .

On the other hand, the well-poised series

'

a, q va, - q if.a, b1 , ... , b.,,; q; z J P+3<DP+2

l.va, - ifa, aq/b1 , ... , aq/bv

"'ill be abbreviated by

P" 2Wr+2 [a; h1 ,. · ·, bv; q; zl.

F. H. Jackson studied the concept .of the basic number at length.

He gave the basic analogues of most cf the specialsurnrnationtheorerns.

In particular. he proved the basic analogues of Dougall'~ theorem as

early as 1909. But he did not give the q-analogues of Watson's and

Whipple's sIJmmation formulas. Andrews [I J used Watson's q-analogue

of Whipple's theorem [10] and obtained the.q-analogues of the termi

nating versions of Watson's and Whipple's summation theorems. We

show here that the q-Watson and q-Whipple formulas of Andrews [I]

are special cases of Jackson's theorem [7; 3. 3. 1. l] for s7. We also

obtain a natural extension of a formula of Verma and Jain [8; 1. 3J in

the form.

12 W11 [a; c;2 q/b2, c, - c, d, - d, abql+m/cd, - abql+m/cd,

q~m, ... q-m, q; q]

[a2 q2; q~]m [h2 q2/c2;q2J.,, [b2 q2/d2; q2]m [a2 q2/c2 d2; q2Jm

[b2 q2; q~·],n [a2 q2/c2; q2Jm [•12 q2fd2; q2],,, [IP q2/c2-d2; q2],,.

• 10W9 [b2; - b2/a, - b2 q/a, a2 q2/b2, c2, d2, a2

b2

b2+2m c2 d2

q-2m; q2; b2 q J a2

'[197

( 1.4)

which is a q-analogue of a known formula [2; 8. 11 J (on replacing a

by -a). In fact, writing the series in reverse order on both the sides of

(I. I) one may get a q-analogue of a well-known formula [2; 8. 1J which will be different in nature from a transformation of Bailey {3].

Similarly, v.e also obtain the q-analogues of the formulas [2.: 8. 21, 8.

31, 8. 41J which are different in nature than the known ones [3J and

[5; 4. 6J. Lastly, as applications of ( 1.1 ), we deduce identities of Rogers

Ramanujan type related to the modulus 23.

2. SPECIALIZATIONS OF JACKSON'S FORMULA

For c = -d =if a, Jackson's summation formula {7; 3. 3. I. IJ

sW7 [a; b, c, d, a2 qI+n/bcd, q-n, q; q]

[aq; qJn (aq/bc; qJn [aq/bd; qJn [aq/cd; qJ,. - [aq/b; qJn raq/c; <tJn [aq/d; q]n [aq/bcd; q]n

reduces to

(2.1)

4<!>

3 · ' ' ' = [-q; q]n [aq; qJn [aq2Jb2; q2]71

[

a. b, - aql+n/b q-n· q· q J aq/b, - bq-n, aql+n [-q/b;q]n [aq2; q2]n ;[aq/b;q]n

(2.2)

(2.2) is a q-analogue of the terminating version of Dixon's theorem

(7; 2. 3. 3. 6J.

< Transforming the 4<!>3 on the left-hand side of (2.2) by the

formula [6J

198]

ra, b, c, q-n; q; q J 4©3 I =

'-d' e, f [e/c; q]n [de/ab;q]n [e; q],. [de/abc; q]:;--

[

d/a, d/b, c, q-n; q; q .1 4$3 J cqI-n cqI-n ..

d, --,--e f

(2.3)

where abcqI-n = def (with d-? aqI+n, e-? aq/b, f - bq-n, c-? b,

b_ -. -aqI+"/b) we get

4©

3 ' ' ' - ' ' = [aq; q]n [aq2/b2; q2Jn

[

q-n ql+n b b· q· q J aql+n, -q, b2 q-n;a [aq2; q2]n [aq/b2; q]~

(2.4)

(2.4) is q-Whipple's theorem due to Ar..drews [l].

Next, transforming the 4©3 on the left-hand side of (2.2) by (2.3)

(with a - -aql+n/b, c-? a, d - aq/b, e - aql+n, f- -bq-11) we get a

kr.own formula [4; 3. 17]. On the other hand transforming the 4©3

(with b - aql+n/b, c -? b, d - aq/b, e -? aql+n, f - bq-11) one may

obtain another known formula [4; 3. 19].

Now, in the non-terminating version of (2.1) i.e. [7; p. 248(15)),

setting c = -d = if a, we get a non-terminating version of (2.2) in

the form

[

a, b, '-aq/bf, f; q; q] [a, f, b/a, bq/f, -aq/bf, ~b2f/a; q] ~ +TI .

aq/b, -bf, aq/f a/b, aq/f, bf/a, b2/a,-bf, -q/f

" . '[b2/a,b, -q/f, bf/a; q; q J _ • 4ID3 -

bq/a, -b2f/a, bq/f

. [b/a, ~b, -q; q · J [b2/2/a, aq2/f2, aq; q2 J =TI TI (2.5)

aq/f, bf/a, -bf. -qlf b2/a

[199

(2. 5) for a = q-" reduces to

[

q-11, b, -qI-11/bj, j; q; q -J 4<l>3 ... qI-11/b;'-bf, qI-njf .

[bf,· q],. (-f)" qn<11-lll2 ., [b; q]n [f; q-]P-•

·. II rb2, b2f2qt., q27-n/J2, i:p-n; q2 ']

Lq, b2f2, q2/f2, b'.'q11 . · (2.6)

Transforming the 4<1>3 on the left hand side by (2.3) (with

d-+ -ql:.;_njbf, c'-+ f, d ~ qi-11/b, e-+ -bf, f-+ r;'I'-n/f) we get q-Watson

s summation formula due to Andrews [ l ]. ·

by

~. PROOF OF (1. 4)

Multiply to the following form of (2. l)

sW7 [a; a2q/b2, bq~, -bq"; -q-'[', q-n; q; q]

b2n [a2q2; q2],. [a2q2jb2; q2] 71 f-:-b2/a; qfan (qa2)" [-aq; qfon (b4/a2; q2]11 [b2/a2; q2]n

[b2; q2],. [b2/a2; q2],. (c2; q2],. [d2; q2],. [b2q2; q4]n [q-2m; q2],. 1

(3;1)

[q2; q2],. [a2q2;q2],. [b2q2/c2; q2],. [b2q2/d2; q2],. (h2; q4],. · •

[a2b2q2+2'."/c2d2; q2]11 q2n [c2d2q-2mja2;-q~]11 [b2q2+2m; q2J,,

and summing with respect to n from o.to m, :we get

10w9 [b2; -b2/a, -b2q/a, a2q2/b2, c2, d2, a2b2q2+2m/c2d2, q-2m; q2;b2q/a2]

m - ~

r=O

{a; q], [aq2; q2]r•{a2q/ b2; q],. [b2q2; q2]2r ·[c2; q2]7

[q; qJ, [a; q2]r [b2/a; q], [a2q2; q2] 2,. [b2q2J.c2; q2],

200]

[d2; q2]r [a2b2q2q2+2mJc2d2; q2Jr [q-2m; q2]r qr b2r [b2q2/d2; q2j~[c!?J2q-2m;a2; q2J-;b2q2+2m; q2], a2r •

. 8w7 [b2q4•; b2/a2, c2q2r, d2q2r, a2b2q2+2m+2•/c2d2, q-2m+2•; q2; q2]

on summing the inner 8W7 by (2. l ), we have (1.1) on some simplifiila-

tion.

In (I.I) setting c2 = -aq2, d = -aq and then summing the resul

ting 6<1>5 ou the right hand side, we .get (on replacing b by bvq)

(I = v-1)

10W9 [a; a2/b2, iqv a, -iqv a, bqm, -bqm', -q-m, q-m; q;q]

[a2q2; q2]m [a2/b2; q2]m (-b2q/a; q]2m b2m [b4q2/a2; q2]m [b2/a2; q2],,. [-a; qfam a2m qm

(3.2) is a q-analogue of the formula [2; § 8 (B)].

Next, we prove the following three transformations

(3.2)

12W11 [a; a2/b2, c, -c, d, -d, abql+m/cd, -abql+m/cd, -q-m, q-m; q; q2]

fa2b2; q2]m [b2q2/c2; q2]m [b2q2/d2; q2]m [a2q2/c2d2; q2]m [b2q2; q2]m [a2q2/c2; q2]m [a2q2/d2; q2j;;;(b2q2/c2d2; q2]m

• 10W9 [b2; a2/b2, -b2/a, -b2q/a, c2, d2, a2b2q2+2m/c2d2, q-2m; q2; b2 q3] a2

14 W1a [a; a2jb2, iqva, -iqva, c, -c, d, -d, abql+m/cd, -abql+m/cd, -q-·m, q-~; q;q]

(3.3)

-1

[201

[a2q2; q2]m [b2q2/c2; q2]m [b2q2/d2; q2]m [a2q2/c2d2; q2]m [b2q2; q2],,. [a2q2/c2; q2Jm [a2q2/d2; q2]m [b2q2/c2d2; q2],,.

10w9 [b2; -b2q/a, -b2q2/a, a2/b2, c2, d2, a2b2q2+2m/c2d2, q-2m; q2;b2q/a2]

(3.4)

and

14 W13 [a; a2/b2q, iqifa, -iqifa, c, -c, d, -d, abql+""/cd, -abql+m/cd,

-q-m, q-rn; q; q2]

[a2q2; q2]m (b2q2/c2; q2]m [b2q2/d2; q2]2m [a2q2/c2d2; q2]m [b2q2; q2]m [a2q2/c2; q£]m [a2q2jd2; q2]m [b2q2/c2d2; q2]m

. 10 w9 [b2; a2jb2q2, -b2q/a, -b2q2/a,c2, d2, a2b2q2+2mjc2d2, q-2m;

q2· b2q2 'Q2]

Proof of {3.3) : We have a summation formula [9; 3. I]

sW7 [a; a2/b2, bq", -'bq", -q-", q-"; q; q2]

[a2q2; q2],. [a2/b2; q2],. [-b2/a; q]2,. b2n q" [b2/a2; q2] 11 [b4q2/a2; q2],. [-aq; qfan a2n

(3.5)

(3.6)

Proof of (3.3) follows on the lines of the proof of ( 1.1) on using (3.6)

instead of ( 4.1 ).

However, from (3.3) one may obtain the summation formula

10W9 [a; a2/b2q, iqif a, -iqif a, bq", -bq", -bq", -q-11 , q-n; q; q2)

202]

[a2q2; q2],. [a2/b2q2; q2],. [-b2q/a; qfo,. b2" q"

[b2/a2; q2]71 [b4q4ja2; q2],. [-a; qfa,. a2" (3.7)

Proofs of (3.4) and (3.5) follow on the lines of the proof of (1.1)

on using (3.2) and (3.7) respectively instead of (3.1).

(3.3)-(3.5) are q-analogues of the formulae of Bailey [2; 8. 31,

8. 21, 8. 41].

4. FURTHER TRANSFORMATIONS

We begin this section by proving the following transformations:

00

4] ~ [a4q4; q oo n, m, r=O

4(n+m +r)2+4(m+r)2+2r2-4p(n+m+r) 4n t8m+ lOr q a [q4; q4],. [q4; q4],;; [q2; q2]r r-aq; qhr [-a2q2; q2fam+2r

~ ); (q-4P; q4]; (-)1 a41 q2iU+ll j=O [q4; q4]1

~ [a; q]n ( l-aq2")n (-)" a.11"

n=O [q; q]n (1-a) - -

and

l(23n- l-8p + l 6j)/2 (5.1)

00

[a4q4; q4]00

.E n, m, r=O

q4(n+m+r)2 + 4(m+r)2 + 2,.2 - 4f(n+m-+ r) + 4(n+2m+2r)

riJ4; q4]n [q4; q4]m [q2; q2]~ [-aq; qJ;----··---

4n+8m+ 10r a

. -[---a2~q-=2,.....; -q~2]_2_m_+-2•

(203

~ f [q-4r-; q4]; (-)i a4j q2i<J+3>. c; [aq; q] .. _(1-a4 qSn+4>

I= 0 [q4; q4]J n=O [q; q]n

lln ( )n n(23n + 15 - 8p + 16j)/2 • a - q (4.2)

Proof of (4.1) (1.1) for d2 = -q-2m, b ~ oo, c ~ oo yields

m ~ [a; q]r (1-aq2r) (-)' a3r qr<7r-ll /2

r=O [q; q], (I-a) [a4q4; q4]m+r [q4; q4]m_r

1 m a2r q2r2

[-a2q2; q2]2, r:O [q2; q2], [-aq; q]zr [q4; q4]m_r (4.3)

Multiply to (4.3) by [x; q4]m [y; q4]m [q-4n; q4]m q4m

[xyq 4nja4; q4]m and sum with

respect tom from 0 to n, we get on using q-analogue of Saalschutz

summation theorem [7; 3. 3. 2. 2] and then letting x, y ~ oo:

n ~

r=O

[a; q]r (l-aq2') (-)" a7r qr<l5r-VJ2 [q; q], (I-a) [a4q4; q4]n+r fq4; q4]n_r

~ 4m+6r 4(m+r)2 + 2r2 ~~-~a __ q·~---~

r, m ;> 0 L42; q2], [q4; q4]m [-aq; q}zr [-a2q2; q21zm+2r

[c/i;q4]n_m:; (4.4)

Using (4 4) in the formula (8, 2 15]

[aq; q]00

~ an qn<n-P> ~n = £ [q-v; q]; (-·a)i qj{J+l>/2

n=O j=O [q; q];

204]

where

00

2: a"' qri<ri-P+2n ot,.

n=O

Gr n

~ .. = 2: r=O [q; q]n_r [aq; q]n+r

for evaluating < ~ .. > with a and q replaced by a4, q4 respectively and

ar = (a; q]r (1-aq2r) (-)" a7r qr<15r-V/2

[q; q]r (1-a) '

we get (4 1).

Proofof(4.2): However (4.3) may be written in the form (see [8]

for details)

t [aq; q_]r (l-a4 q8r+4) (-)' a3r qr<7r-Vt2

r=O [q; q], [q4; q4]n,_, [a4 q8; q4]m+r

(l-a4q4) [-a2q2; q2]2,,,

m ~ 2~ 2: a q

r= 0 [q2; q2]r [-aq; qfor [q4; q4],,,_r (4.5)

Proof of (4.2) follov\S on the lines of the proof of (4.1) on using (4.5)

instead of (4.3).

Identities of Roger S-Ramanujan Type Related to Modulus 23

(4.1) for p = 0, a = I, q. q2, q3, q4 and q5 yields the following identities

[q4;q4] oo q4(n+m+r)2 + 4(m+r)2 + 2r. 2 ~--ex: ~ [q; q]oo n, m, r=O [q4; q4], [q4; q4]m [q2; q2]r [-q; qfa,. [-q2; qfam+2r

= 11 (l-q")-1 n ~ 0, 11, 12 (mod 23) (4.6)

(205

[q4; q4J00 ~

[q; qJ 00 .n,m, r=O

q4(n+m +r)2+4(m+r)2+2r2+4n+Sm+ lOr

[q4; q4j 11 {q4; q4Jm [q2; q2], [-q; qJ;;.-+l F-1}2; q2]2m+2r+1

TI (l-q")-1 = n ~ 0, I, 22 (mod 23) (4.7)

[q4; q4J oo 4(n+m+r)2 + 4(m+r)2 + 2r2 + 2r ~~-oo- E ~q-.-.-::----;---c.-----c=--=-=-,,-__,,~--=--,,-~ [q; q]oo n m r=O [q4;q4],. [q4 ;q4]m [q2;q2], [-q;qJ2r [-q2;q2fam+2r

' '

[q4; q4J00

[q; qJOO

_ TI (I-q")-1 - n =/= 0, 10, 13 (mod 23) '

00 E

n, m, r = 0

q(n+m+r)2 + 4(m+r)2 + 2r2+4.n+Sm+ IOr

[q4; q4]n [q4;-q4]m (q2; q::?Jr {-q; qfad-q2; q2]2m+2r+l

[q4; q4]00

[q; q]oo

_ TI (I - qn)-1 - n ~ 0, 2, 21 (mod 23)

00 ~

n, m, r = 0

q4(n+m+r)2 + 4(m+r)2 + 2r2 + 20r+Sn+16m+9

[q4; q4]n [q4; q4]m [q2; 92jr [-q; qfor+i [-q2; q2]2m+2•+2

(4.8)

(4.9)

=TI (l-q")-1 n~O, 10, 13 (mod 23)

= TI (l-q")-1 n,;\:O, 9, 14 (mod 23) (4.10)

[q4; q4J00

[q; q]oo

00 ~

n, m, r = 0

206J

q4(n+m+r)2+4(m+r)2+ 2r2+8m+4n+ 12r(I +q2+q~r+l) [q4; q4Jn [q4; q4Jm (q2; q2J• [-q; qJ2r+I [-q2; q2fam+2'+1

= Il (l-qnr1 11 ~ 0, 3, 20 (mod 23) (4. ll)

Next, (4.2J for a= I and p =0, I gives

[q4; q4J=

{q; q]=

= ~

n, m, r = 0

q4(n+m+r)2 + 4(m+r)2 + 2r2+4n+8m+8r

'!: .· -[q4; q4]n [q4; q4]m (q2; q2J, [-q; q)2, [-q2; q2fo_m_+_2,-

= IT { l -'--'qn)-1 n ~ 0, 4, 19 (mod 23)

[q4; q4J=

[q; qJ=

= ~

n, m, r = 0

q4(n+m+r)2 + A(m+r)2 + 2r2+4m+4r -----[q4; q4Jn [q4; q4]rn [q2; q2Jr [-q; qfo, [-q2; q2]2m+2r

= IT (1-q")-I 11 ~ 0, 8, 15 (mod 23)

(4.12)

(4.13)

Furthermore ( 4 .1) for a = 1, p = 1 gives the following identity

(on using (4 12))

[q4;q4]=

[q; qj: = ~

n, m, r = 0

4(n-Pm-tr)il + 4(m+r)2 +,2,.2 .- 4~n+m+r) .(l 4n+8m-j-8r) q ·-~q

[q4; q4], [q4; q4],,, [q2; q2], [-q; 4Jzr [-q2; q2fam+2r

= n (l-qn)-1 n ~ 0, 7, 16 (mod 23) (4.14)

(4.l)for a =q,p := 1 reduces to

[q4; q4]00

[q; q]oo

00 L

n, m; r = 0

q4(n+m+r)2+ 4(m+r)2 + 2r2+4·n+6r

-[q4; q4] 11 [q4; q4]m (q2; q2] 7 [-q; qfarH [...:q2: q2fa,,,-f2r+1

= IT ( l -q")-1 n ~ 0, 5, 18 (mod 23) -q = IT (l-q")-1

n ~ 0, 3, 20 (mod 23)

['207

( 4.15)

Lastly, ( 4.11) for a = q3, p = l gives the following identity (on

using (4.15))

[q4; q4]00

[q; qjoo

00

L n, m, r = 0

q4.(n+m+r)2 + 4(m+r)2 + 2r2+4m+6r (l+q4m+4n+4r+.2 )

[q4; q4] 71 [q4; q4]m (q2; q2], [-q; q]zr [-:q?.; q2]2m+2•+l

= II (l-q")-1 n ~ 0, 6, 17 (mod 23) (4.16)

Acknowledgements

I would like to express my gratitude to Professor H. M. Srivastava

for his suggestions and for redrafting this paper in its present form.

REFERENCES

[l] G. E. Andrews, On q-analogues of the Watson and Whipple

summations, SIAM J. Math. Anal. 7 (1976), 332-336.

[2] W. N. Bailey, Some identities involving generalized hypergeom

etric series, Proc. London Math. Soc. (2) 29 (1929), 503-516.

[3] W. N. Bailey, A transformation of nearly-poised basic hyperge-

208]

ometric series, J. London Math. Soc 22 (1947), 237-240.

(4} V. K. Jain, Some transformations of basic hypergeometric func

tions. II. SIAM J. Math. Anal.12 (1981), 253-257:

(5] V. K. Jain, Certain transformations of basic hypergeometric

series and their applications, Pacific J. Math. 101 (l9fl2),

333-349.

[6J D. B. Sears, On the transformation theory of basic hypergeom

etric functions,. Proc. London Afath So.c. (2) 53 ( 1951 ),

158-180.

[7] L. J. Slater, Generalized Hypergeometric Functions, Cambr~dge

University Press, 1966.

[8] A. Verma and V. K. Jain, Tranformations between basic hyperg

eometric series on different bases and identities of Rogers

Ramanujan type, J. Math. Anal. Appl. 76 (1980),' 230-269,

[9] A. Verma and C. M. Joshi, Some remarks on summation of

basic hypergeometric series, Houston J. Math. 5 (1979),

277-294.

[10] G. N. Watson, A new proof of the Rogers-Ramanujan identities,

J. London Math. Soc. 4 (1929), 4-9~

Jnaniibha, Vol. 15, 1935

UNIFORM SPACE VALUED UNIFORMLY ALMOST PERIODIC

FUNCTIONS DEPENDING ON PARAMETERS

By

SUMAN JAIN and BIJENDRA SINGH

School of Studies in Mathematics, Vikram University,

Ujjain-456 010, M. P., India

l Rece.ived: May 31, 1984; Revised: November 2, 1984)

ABSTRACT

Sharma, Reddy and Funakoshi [3] studied some fundamental

prnperties of uniform space valued almost periodic functions depending

on parameters. The purpose of this paper is to generalize some results

given in { 1 J on uniform spaces. For uniform spaces and the related

nations, see [2]

1. INTRODUCTION AND DEFINITIONS

Let T and E be uniform spaces. We shall study here only, E-valued

functions defined on T x R • where R is the real line, and assume that

any function in theorems or definition is continuous.

Definition 1 : Letf(t, x) be a real or complex function defined for

all real values of x. A number -r is called a U translation number .of

f(t, x) if

U. b. ( f(t, X -j-- -r), f(t, X)) Eu, -=<x<oo

Whenever /ET and nR.

V\ e denote the set of all U tramlation numbers of a function f(t, x) by

E { U, j(t, x)}.

210]

Definition 2 : A continuous functionf(t, x) is called uniformly almost

periodic in x, uniformly with respect to tET, if to each entourage U of

R the set E{ U, f(t, x)} is relatively dense.

2. MAIN RESULTS

We first state and prove

Theorem 1. If T is a compact uniform space and f(t, x) is an unifo

rmly almost periodic function in x uniformly with respect to tET, irhen

the set {f(t, x)/tET, XER} is precompact.

Proof; Let V be an entourage of E. Then there is a symmetric ento

urage U of E with u2c V. Let (t0, xo) be an arbitrary e'.ement of Tx R.

By the definition of uniformly almost periodicity of f(t, x), there is a

U translation number 't' of the set E { U, f(t0 , x0)}. Obviously, the set

s = {f(t, .\)ftET, XE [O, k (U)]}

where k(U) is a positive number, is compact. So there are a1 , ... , an fl

ES= U U {a1) for which we can find an m E {l,. .. ,n} such that i=l

f(to, xo + 't') E U (am), since x 0 + T E [O, k(u)],

or, equiv~lently,

(f(to, xo + T), am)€ U.

Also, since (f(to, xo + 't'), f(to, xo))E U,

it follows from the above that

(f(t0 , x 0 ), a,,.) E U2 CI or /(to, xo)" V(au,).

Therefore, v.e have

{f(t, x)jtET,XER} C n

U V(a,) i=l

which proves the above theorem.

Next we state

[211

Tbeorein 2. lfT is a compact uniform space andf(t, x) is an unifo

rmly almost periodic funcMon in x uniformly with respect to tET, then

the function f(t, x) is uniformly continuous on T x R.

Proof. Let V be an entourage of E. Then there exists a symmetric

entourage U of E with uac V. Let k( U) be the length associated with

U such that (f(t1 , y,),f(t2 , Y2)) E U for any (ti, y 1), (t2, Y2) belonging

to the interval T x lO, l + k( U)] if only (!2, ti) E W, I Y2• Yi I < il

where o is a positive number o = o (U) < l. Let now (t1, x1), (t2, x2)

be any two _r,oints of T x R with (t2 , t1) E W, where W is an entourage

of T and I X1> x2 l < o. There exists a U translation number " of the

set £{ U, f(t, x)} such that both the numbers Ct1, xi. + 't'), (t2, x2 + 't') belong to the interval T x. [O, 1 + k ( U)].

we have then

(f(ti, X1 + 't'),j(t2, X2 X 'C')) cJf,

On the other hand

(f(t, x + T), f(t, x)) EU, for any XER.

Thus

(f(t1, X1),f(t2, X2)) EU3 c v.

Therefore, f(t, .x) is uriformly continuous on Tx R~

This proves the theorem.

212)

3. FURTHER RESULTS

Theorem 3. If {fn (t, x)} is a sequence of uniformly almost periodic

functions in x uniformly with respect to tET which is uniformly converg

ent on TxR to a function f(t, x) is also uniformly almost periodic in x

uniformly with respect teT.

Proof. Let V be on entourage of E. Then there exists a symmetric

entourage U of E with U3 C V. Thus we can choose an no such that

(f(t, x), f (t, x)) e U no

for every (t, x) e Tx R. Non let, be a translation number of'

E{ U, f (t, x)}. Then · no

(f(t, x + •), f(t, x)) e U3 C V,

which shows that

£{ U, f(t. x)} :J £{ U, f (t, x)}. . no

Thus E{ U,f(t,x)} is relatively dense. Consequently, f(t, x) is uniformly

almost period.

Theorem 4. The sum of two uniform space valued uniformly almost

periodic functions f 1 (t, x), f 2 (t, x) depending on parameters is again

uniformly almost periodic function depending on parameters.

Proof. Let, be any U translation number of the set

E { U, / 1 (t, x)}, E{ U,f2(t, x)}.

then

(Ji (1, x + •) +f2 (t, x + •) -/1 (t, x) -f2(t, x)) eU,

which shows that 'is a U translation number of the set

[213

E{ U, Ji (t, x) + f 2 (t. x)}. Thus

E{ U, !1 (t, x) + fz (t, x)} :J E{ U.f1 (t, x)}. E{ U, f 2 (t, x)}.

Therefore, E{ U, f 1 (t, x) + f2 (t, x)} is relatively dense. Consequently,

f 1 (t, x) + f2 (t, x) is uniformly almost periodic depending on param

eters, which proves the theorem.

REFERENCES

[l] A. S. Besicovitch, Almost Periodic Functions, Dover Puhl. New

York, 1954.

[2] J. L. Kelley, General Topology, Van Nostrand, New Yolk. 1955.

[3] P. L. Sharma, K. C. A. Reddy and S. Funakoshi, Remarks on

uniform space valued almost periodic functions depending

on parameters, Math. Sem. Notes KObe Univ. 4 (1976),

87-90.

Jfianabha, Vol, 15, 1985

' THE HARMONIC CESARO PRODUCT SUMMABILITY

OF THE DERIVED FOURIER SERIES

By

KALPANA SAXENA

Department of Mathematics, Government College,

Shabdol-484001, M. P., India

( Received: March 5, 1984; Revised [Final] : April 15, 1985)

1. DEFINITIONS AND NOTATIONS

If an denotes the (C, I) Transform of partial sum Sn of an infinite

00

series ~ Un and 0

lim I n (I.I) ~

a,._,, =S,

n-+00 log n k=l -k-

00

where Sis finite, then the series ~ Un is said to be summable (H,l) 0

( C,l) to S, or symbolically

00

~ U,. = S [(H,1) (C,1)] 0

Let f(x) be a function integrable in the sense of Lebesgue over

the interval (- re, re) and periodic with period 2re.. Let the Fourier

series associated with f(x) be

(1.2) 1 2 an 00 00

+ ~ (an cos nx + bn sin nx) = ~. A,. (x) n=l . 0

216]

The derived series of the Fourier series ( l.2) is

00 00

(1.3) L; n (b,. cos nx - a,. sin nx) = L; n B,. (x) n=l 1

We shall use, for fixed x ands the following notations :

cp (t) = f(x + t) + f (x - t) - 2s

g (t) = f ( x + t) + f (x ~ t) 4 sin t/2 __ _

2. INTRODl:IC:TION

Siddiqi [3'} and Hille; ,and< Tamarl.dn. [ l] have, respectively. esta

blished the following theorems on the harmonic summability of the

Fourier series :-

Theorem A (Siddiqi [3]) If

(2.l) <fl(t) = I: I ll>(u) I du= 0 (t/log l/t), as t-+ 0

00

then 2-' An(x) = S(IJ. 1). 0

Theorem B (Hille and Tamarkin [1]). If

(2.2) ©(t) = ft I ll>(u) I du = o (t), as t--+ o, lo

and

(2.3)1"1) rc/n

I «l>(t) - «I>(t+rc/n) I t

log 1/t dt = 0 (logn). as n -+ oo,

where "I) is a pas itive constant, then

00

~ A,,(x) = S(lf, I). 0

[217

Since the derived Fourier series is neither (C, I) summable nor (H, I)

under tbe hypothesis of Theorems A and B, respectively. Therefore it

is natural to expect the extensions of Theorems A and B to the product

summability (H, 1) (C, 1) of the derived Fourier series under analogous

conditions with cfo(t) replaced by g(t), with this point of view, we prove

here

Therefore 1. If

(2.4) G(t) = I: I g(u) I du= o I t I logl/t), as t _,. 0,

then the derived Fourier series {1.3) is summable (H. 1) (C. I) to zero,

at the point x.

Theorem 2. If

(2 5) G(t) = 1: I g(u) I du= o(t), as t-+ 0,

aud

(2.6) J g(t) - g(t+rc/n) I J

'fJ

rc/n t log l/t dt = o(logn), as n -+ oo,

Where 'fJ, is a positive constant, then the derived Fourier Series (1.3) is

summable (H. I) (C. l) to zero, at the point x.

3. Proof of Theorem 1.

Following Sachan (2] the ( C, I) transform { crn+i}

218]

of the sequence { Sn+i} of partial sums of the derived Fourier series is

given by

crn+l = ~ ra g(t) [ 2{1-cos(n+ l)t} -1..

TC J O (n +I )t2 - ' sin(n+l)t

(n+ I)t

1 sin (n + 1 )t j dt + o(I)

t J

We choose B > 0 so .small that fort c· {O, o) the cond.ition (2.3) is sat

isfied.

Denoting the harmonic transform of crn+i. i.e., the(H. 1), (C. I)

transform of Sn+I by Hn, we have, by the regularity of the method of

summation,

Ia n

Hn = ___:____ . g(t) }.; TC!ogn 0 r=l (n-r+l)

[ 2(1-cos rt)

rt2 + sin rt

rt _ !i; rt ] dt + 0(1)

2 Jrc/n n 1 --- g(t) }.; -

TC logn 0 r= I (n-r+ l)

[ 2(I~cos rt) +

rt2 sin rt _ sin rt l dt

rt t J

+ __ g(t) }.; __ I 2(1-cos rt) 2 Ja n [ TC /ogn rr./n r= 1 (n-r+ 1 rt2

2 re !ogn J

a n 'g(t) ~

TC/n r= 1

sin rt (n-r+I)t dt + o(t)

+ sin rt Jdt rt

. [219

(3.1) = 2/rc [P + Q-R] + 0(\), say

Since for t > 0,

2(I~cos rt) + s_in rt rt2 rt

sin tr -· t = O(r) 4

We obtain

I P I = _I_ ~ I J'TC/n logn r= 1 (n-r+ 1)

0 J g(t) I O(r) dt

I ITC/n -- O(n logn) I g(t) I dt (logn . 0

(3.2) = o(I), as n-+ oo.

By an integration by parts, we obtain

J QI= __ I I ~- l {3

g(t)l2(l-cosrt) + sin rt ld \ Jogn r=l r(n+l-r) )'TC/n L t2 - 1-J /

= o ( ___ l - ). £ ) 1/r + 1 } fa I g(t) I dt (n+l)logn. ,.~1l (n+l-r) .. Jn:/n t2

(3.3) = o(l ), as n -+ oo.

Since for t E(7t/n, a),

n ~ _ sin rt I (n-r+l) = O(log 1/t)

By an integration by parts, we get

l RI= o (-1-) f3

I g(t) I (log l/t) dt logn j'TC/n t

220]

= O (-1-[l G(t) log 1/t }3

.+ f 3

G(t) (~+lo-L!l.£_ )dtl logn l t n/n Jn/n t2 J

= O (-,-1 -)llo(J) + o(l) Hi) ( /

1 + 1/t )dt J ogn Un/n t og 1/t

( I [ · 1 = o(I) + 0 -- o(log logn) + o(logn) J

logn

(3.4) = o(l), as n _,.. oo.

Finally, from (3. J), (3.2), (3,3) and (3.4). we find

H .. = o(l), as n-+ oo

This completes the proof of Theorem I.

4. Proof of Theorem 2

It may be noted that from the proof of Theorem I that P, Q = o(I)

under the hypothesis (2.5). Thus to prove Theorem 2, it is sufficient to

show that R = o(I), under the hypothesis (2.5) and (2.6)

We write

R = ~I- r 3 g(t) . n

!ogn jn/n --t- stn(n+l)t ~ cosrt dt r= 1 r

!a -- g(t) n logn n/n -

1- cos (n+ I) t ~ sin rt dt

r=l r

= R1 - R2, say

(2.6) and an integration by parts give

""';D:

1 fa I g(t) ! 0(1) dt I R2 I= logn Jrt/n t

= o(I), asfor (3.3).

Further, we write

2R1 = __ 2 _ f S g(t) sin (n+ 1 )t log (l /2 sin t /2) dt logn j rt/n t

_2_ rs g(t) sin nt log 1/t dt + o(l) logn J rt/n t

1 ris-rt/n JS 'j/g(t)\ logl/tsinntdt = -L + t logn rc/n S-rt/n

[221

1 [frc/n rs-rc/n] g(t+rt/n) log(lft+rr;/n)sinntdt+o(l) - logn Jo + Jrt/n (t+rc/n)

1 logn

rs-rt/n -~~-g(:+rtfn) log (l/t) sin nt dt

J rc/n

+ __ . rs-rt/n g(t+rt/n)[log 1/t - log lit+ 'TC/n ]sin ntdt logn Jrt/n . t

1 l S-TC/n r J -.J. + -- g(t + 7t/n) j l/t - -(·---- log(lft+rtfn)sin nt dt logn rc/n L t+'TC/n)

1 r'TC/n g(t+rt/n) log (I/t + 'TC/n) sin nt dt - logn Jo (t+'TC/n)

+ _!__rs g(t) log ljt sin nt dt + o(l) logn h-'TC/n t

222]

= Li + L2 + L3 ~ L4 + L5 + o(l)

By (2.6), we get

I L1 I = o(I), as n-+ oo

An integration by parts, we have

1 1'8-rc/n I g(t+rc/n) I log O+rcfnt)dt I L2 I = logn rc/n t

= o (.-1_) ·1a-rc/n I g(t+rc/n) I dt log n rc/n

= o(l), as n-+ oo.

Again, on integration by parts, we obtain

I L3 I < . I g(t+rcfn) I -~ rc/n . dt · "I'il-rc/n . .

rc/n t(t+rtfn)

= 0(1/n) I g(t+'rt/n) I dt J

'a-rcin . . ..

rc/n t

[{ . :}8-rc/h :J'a-rc/n J

= 0 (1/n) o(t+rc/n) l/t2 . + o(t+rc/n)l/t3 dt rc/n rc/n

= 0(1/nf{o(l/t)'}a-rc/n + 'I''8~1t/n o(l/i2) dt J L rc/n rc/n

= o(l), as n-+ oo.

Similarly, we fined

.1 , J2rc/n . ! g(~)· j fog 1/ t dt I L4 I ~ log n rc/n

[223

12 rc/n

= o (n) I g(t) I dt n;/n

= o(l), as n -> oo

and that

j fog (3-rc/n) l I g(t) I dt !a

i L5 I ~ (3-TC/n) log n '6-r../n

= o( I ) , as n -+ oo

Thus, collecting the results, we obtain

R = oO), as n -+ oo,

and the proof of Theorem 2 is completed.

Acknowledgements

The authoress is extremely grateful to Dr. P. D. Katha! and Prof. M. P. Schan for their kind interest and valuable suggestions in the perparation of this paper. She is also thankful to prof. H. M. Srivastava for his generous encouragement and very special suggestions for the improvement of the paper.

REFERENCES

[I] E. Hille and J. D. Tamarkin, On the summability of Fourier

series. I, Trans. Amer. Math. Soc. 34. (1932), 757-783.

[2] M. P. Sachan, Matrix. summability of the derived Fourier series,

Jfianabha 13(1983), 47-54.

[3] J. A. Siddiqi, On the harmonic summability of Fourier series,

Proc. Indian Acad. Sci. Sect. A 28 {1948), 527-531.

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CONTENTS

Time Dependent Slow· Flow of A Viscous Incompressible Fluid between Wavy Walls -D. ,\!l. Mehta & r. N. Gaur

Relations between The Hankel & Multi variable H. Function Operators -r. S. Kumara Swamy& S. N. Mathur· 19.

Fixed Point Theorem For .Mapping with A Generalized Contractive Iterate, in 2-Met;ric spaces V. H. Badshah & Bijendra Singh 27

A Fixed.Point Theorem for·L-Contractions in Generalized Metric spaces. -A Carbone 33 ·

Integrals Involving A General Class of Polynomials

Some Results cin Fixed Points For Three Maps

-A~ K. Arora &~ C. L. Koul 39

-As!iok Ganguly 51

A Note .on Entire Functions of Irregular (p, q)- Growth -G. S. Srivastava & H. S, l(asana, 55

Applications of Srivastava's is Hypergeometric Functions of Three Variables' in· Heat Conduction -R. C. Singh Chandel and B. N. Dwivedi 65 ·

Integrals Involving The H- Function. -Sujata Verma 71

On Fixed Points of weakly Commuting Mappings, in. Compact Metric.spac;es •. -,-Salvator~ Ses~a & Brian Fisher 79

Remarks on Some Fixed Point Theorems & .Their Extensions •.. -D. E. Anderso.n, M. D. Gua_v &.K. L. Singh· .93

On I v, ,\ 11, Summability of Ultraspherical Series - W. T. Sulaiman I 03

On Strong Approximation to A Functiori 'by it~ jac~bi Ser.ie~ '.:;;"'::'.

-W. '.f. S,.~laiman 121

A Note On A cla~s of Analytic Functions in the .. Unit pisk .·. I

A Note On the R!adii of Starlikeness & Convexity.

, -Sl/igeyoshi Owa 127

'-'Sk'igecjoshi Ow~ l S l Some Remarks On Common Fixed Points of Four ·Mappings; ·' ·: " v. .

',\"·

-M. L. Diviccaro &. s: Sessa IS9

Some Fixed Point Theorems For Pairs 'of wlappings, co :: .,,,.B. E. Rhoades .1'51 ' • ·•. . ;:· . :• ;-;•,: ~I ·> "' . ~···· . '

Effect of Viscosity on Rayleigh,-Taylor Instability. in T~e Presence. of 'A · - ";'"·

Vertical Magnetic Field . . -B,. M. Sharma ~J'.57··'

Some Integrals and Series of The Product of Two Multi variable fl;;.Functiorts.

· , -r. N. Prasad&· K. Nath 169

Finite Integrals Involving the Products of the H-Functions of Two and Mo~~ Variables ~R. M. Jain 183

Certain. Applications of Jackson's Summation Formula -V. K. Jain 195

Uniform Space Valued Uniformly Almost Periodic Functions Depending on

Parameters . -Suman Jain and Bijendra Singh 209

The. H.armonic Cesaro Product Summability of the Derived Fourier Series

'---Kalpana'Saxena 215

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