P=:ooodownloads.hindawi.com/journals/ijmms/1998/751248.pdfLet Eam be an infinite series with partial sums sn Let anandrb denotethe nth Ceshro meanof order ,5(6 > 1) ofthesequences

Internat. J. Math. & Math. Sci.VOL. 21 NO. 3 (1998) 603-606

603

ON A NEW ABSOLUTE SUMMABILITY METHOD

W.T. SULAIMAN

Department ofMathematicsCollege of ScienceUniversity of QatarP.O. Box 2713Doha, QATAR

(Received July 1993 and in revised form March 21, 1997)

ABSTRACT. A theorem concerning some new absolute summability method is proved. Many other

results, some ofthem known, are deduced.

KEY WORDS AND PHRASES: Absolute summability.1991 AMS SUBJECT CLASSIFICATION CODES: 40C.

1. INTRODUCTIONandLet E am be an infinite series with partial sums sn Let an rb denote the nth Ceshro mean of

order ,5(6 > 1) ofthe sequences { s,, } and {na} respectively. The series E o is said to be summable

[C, 6[k, k >_ 1, if

kk-1 [grn O.n_ < oo,

or equivalently

n--]

Let {p, } be a sequence of positive numbers such that

P=:ooo as --,oo(P_,=r,_,=O,i>_ 1.v--O

The series Ian is said to be summable I,11, k _> 1, if(Bor [1]),

(P./r)-Xlt. t_l < co,

whe

The series : an is said to be summable IR, Plk, k _> 1, ifBor [2],

E nk-lltn tn-llk < 00,n--1

604 W.T. SULAIMAN

If we take p, 1, each of the two summabilities [-,Pik and [R,p, lk is the same as

summabililty Let {} be any sequence of positive numbers. The series E a is said to be summable

I-,p,lk, k >_ 1, if(Sulaiman [31),

Cnk-1 It, t_ < oo.

It is clear that

We assume {a,}, {} and {q} be sequences ofpositive numbers such that

Q ,q oo.v--0

We prove the following.THEOREM 1. Let t. denote the C,p)-mean of the series Ea and write To -l/At-l.

(I)

and

then the series E a e. is summable l, q,,, c.l k _> 1.

2. PROOF OF THEOREM 1

Let -. be the (, q,)-mean ofthe series E a, e,. Then

AT.n_ qnQ,.,Q,_,

Q,,-lave.vv=l

Tn,l -I- Tn,2 -- Tn,3 -I- Tn,4, say.

To prove the theorem, by Minkowski’s inequality, it is sufficient to show that

NEW ABSOLUTE SUMMABILITY METHOD 605

a. < oo, r-- 1,2,3,4.n--1

Applying HOlder’s inequality, with indices k and k’, where + - 1, we have

k

k-I Ik E k-1 qn Qv. l,:ll/k-lT,.,=2 .=2 Q.Q.- .= q

-1 q Q_

__1<n=2

an Q:I=I

qolel N IT.I Qn_lv=

0(1)1 qv /ql’lSl-klnl

n=v+l

k-1 qn

()_10(1) lellTl.v=l

I(-)()n=l n=l

()-’(_) ( )_< o(1) " q I,,,IIT,,In=l

3. APPLICATIONSTHEOREM 2. If

,. 0(), e.q. 0(rO.),

and (I) is satisfied, then the series E a. is summable I, qn, an [, whenever it is summable

PROOF. Follows from Theorem by putting en 1.

COROLLARY (Bor [1] and [4]). If

606 W.T. SULAIM

,q O(Q), Q o(,q),

then the series E a, is summable I, q’[k iffit is summable IC, 1[ k, k _> 1

PROOF. Applying Theorem 2 with a, Q,/q, fl, n, and i 1. We have a, 0(fl,),Pnq, O(p,Q,), and (I) is satisfied. Therefore ]C, lib = I-,q,[k. Now the same application of

Theorem 2 with a, n, , P,/p,, we obtain the other way round.

COROLLARY 2 (Bor and Thorpe [5]). If

P,.,q,., O(p,.,Q,.,), p,.,Q, O(P,q,) (II)

then the series E a,, is summable [, qlk iffit is summable I, P,’,lk, k >_ 1.

PROOF. Applying Theorem 2 with a, Q,.,/q,.,, lg, P,,/p,.,. Clearly an 0(/) and (I) is

satisfied. Therefore I,p[k I, qnlk" The result is still valid ifwe interchange {p,} and

COROLLARY 3. Suppose that (I) is satisfied for p and q, (II) is also satisfied and that {%/p,) is

nonincreasing, then the series E a is summable [R, q[k iffit is summable [R, Plk, k _> 1.

PROOF. Applying Theorem 2 with a, , n. It is clear that [R, p,[ = ]R, qlk. For the

other direction, it needs to be shown that (I) is satisfied if we are replacing q,, by p Since {%/p} is

nonincreasing, we have, using (II),_k-l_k

n=v+ PnkPn-1

It may be mentioned that Corollary 3 gives an alternative proof to the sufficiency part of the theorem in

[2].

REFERENCES

[1] BOlL H., On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), 147-149

[2] BOR, H., On the relative strength of two absolute summability methods, Proc. Amer. Math. Soc.113 (1991), 1009-1012.

[3] SULAIMAN, W T., On some summability factors of infinite series, Proc. Amer. Math. Soc. 115(1992), 313-317.

[4] BOR, H., A note on two summability methods, Proc. Amer. Math. Soc. 911 (1986), 81-84.

[5] BOIL, H. and THORPE, B., On some absolute summability methods, Analysis 7 (1987), 145-152

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P=:ooodownloads.hindawi.com/journals/ijmms/1998/751248.pdfLet Eam be an infinite series with partial sums sn Let anandrb denotethe nth Ceshro meanof order ,5(6 > 1) ofthesequences