Research Article Generalized Growth of Special Monogenic ...downloads.hindawi.com/archive/2014/510232.pdf · Research Article Generalized Growth of Special Monogenic Functions SusheelKumar

Research ArticleGeneralized Growth of Special Monogenic Functions

Susheel Kumar

Department of Mathematics Jaypee University of Information Technology Samirpur 177601 (HP) India

Correspondence should be addressed to Susheel Kumar sus83dmagmailcom

Received 6 January 2014 Accepted 2 March 2014 Published 1 April 2014

Academic Editor Yan Xu

Copyright copy 2014 Susheel Kumar This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We study the generalized growth of special monogenic functions The characterizations of generalized order generalized lowerorder generalized type and generalized lower type of special monogenic functions have been obtained in terms of their Taylorrsquosseries coefficients

1 Introduction

Clifford analysis offers the possibility of generalizing complexfunction theory to higher dimensions It considers Cliffordalgebra valued functions that are defined in open subsets ofR119899 for arbitrary finite 119899 isin N and that are solutions of higher-dimensionalCauchy-Riemann systemsThese are often calledClifford holomorphic or monogenic functions

In order to make calculations more concise we use thefollowing notations where m = (119898

1 119898

119899) isin N119899

0is 119899-

dimensional multi-index and x isin R119899

xm = 11990911989811 119909

119898119899

119899

m = 1198981 119898

119899

|m| = 1198981+ sdot sdot sdot + 119898

119899

(1)

Following Almeida and Krauszlighar [1] and Constales et al [23] we give some definitions and associated properties

By 1198901 1198902 119890

119899 we denote the canonical basis of the

Euclidean vector space R119899 The associated real Cliffordalgebra Cl

0119899is the free algebra generated by R119899 modulo

x2 = minus||x||21198900 where 119890

0is the neutral element with respect

to multiplication of the Clifford algebra Cl0119899 In the Clifford

algebra Cl0119899 the following multiplication rule holds

119890119894119890119895+ 119890119895119890119894= minus2120575

1198941198951198900 119894 119895 = 1 2 119899 (2)

where 120575119894119895is Kronecker symbol A basis for Clifford algebra

Cl0119899

is given by the set 119890119860 119860 sube 1 2 119899 with 119890

119860=

1198901198971

1198901198972

119890119897119903

where 1 le 1198971lt 1198972lt sdot sdot sdot lt 119897

119903le 119899 119890

120601= 1198900= 1

Each 119886 isin Cl0119899

can be written in the form 119886 = sum119860119886119860119890119860

with 119886119860isin R The conjugation in Clifford algebra Cl

0119899is

defined by 119886 = sum119860119886119860119890119860 where 119890

119860= 119890119897119903

119890119897119903minus1

1198901198971

and119890119895= minus119890119895for 119895 = 1 2 119899 119890

0= 1198900= 1 The linear subspace

span1198771 1198901 119890

119899 = R oplus R119899 sub Cl

0119899is the so-called space

of paravectors 119911 = 1199090+ 11990911198901+ 11990921198902+ sdot sdot sdot + 119909

119899119890119899which we

simply identify withR119899+1 Here 1199090= Sc(119911) is scalar part and

x = 11990911198901+11990921198902+sdot sdot sdot+119909

119899119890119899= Vec(119911) is vector part of paravector

119911 The Clifford norm of an arbitrary 119886 = sum119860119886119860119890119860is given by

119886 = (sum

119860

1003816100381610038161003816119886119860

1003816100381610038161003816

2

)

12

(3)

Also for 119887 isin Cl0119899 we have 119886119887 le 2

1198992

119886119887 Eachparavector 119911 isin R119899+1 0 has an inverse element in R119899+1

which can be represented in the form 119911minus1

= 119911||119911||2 In

order tomake calculationsmore concise we use the followingnotation

119896m = (119899 + |m| minus 1

|m| ) =

(119899)mm

(119899)m = 119899 (119899 + 1) (119899 + |m| minus 1) (4)

The generalized Cauchy-Riemann operator in R119899+1 is givenby

119863 equiv

120597

1205971199090

+

119899

sum

119894=1

119890119894

120597

120597119909119894

(5)

Hindawi Publishing CorporationJournal of Complex AnalysisVolume 2014 Article ID 510232 5 pageshttpdxdoiorg1011552014510232

2 Journal of Complex Analysis

If 119880 sube R119899+1 is an open set then a function 119892 119880 rarr Cl0119899is

called left (right) monogenic at a point 119911 isin 119880 if 119863119892(119911) = 0(119892119863(119911) = 0) The functions which are left (right) monogenicin the whole space are called left (right) entire monogenicfunctions

Following Abul-Ez and Constales [4] we consider theclass of monogenic polynomials 119901m of degree |m| defined as

119901m (119911) =infin

sum

119894+119895=|m|

((119899 minus 1) 2)119894

119894

((119899 + 1) 2)119895

119895

(119911)119894

119911119895

(6)

Let 119908119899be 119899-dimensional surface area of 119899 + 1-dimensional

unit ball and let 119878119899 be 119899-dimensional sphere Then the classof monogenic polynomials described in (6) satisfies (see [5]pp 1259)

1

119908mint

119878119899

119901m (119911)119901l (119911) 119889119878119911 = 119896m120575|m||l| (7)

Also following Abul-Ez and De Almeida [5] we have

max119911=119903

1003817100381710038171003817119901m (119911)

1003817100381710038171003817= 119896m119903

m (8)

2 Preliminaries

Now following Abul-Ez and De Almeida [5] we give somedefinitions which will be used in the next section

Definition 1 Let Ω be a connected open subset of R119899+1

containing the origin and let 119892(119911) be monogenic in Ω Then119892(119911) is called specialmonogenic inΩ if and only if its Taylorrsquosseries near zero has the form (see [5] pp 1259)

119892 (119911) =

infin

sum

|m|=0119901m (119911) 119888m 119888m isin Cl0119899 (9)

Definition 2 Let 119892(119911) = suminfin|m|=0 119901m(119911)119888m be a special mono-

genic function defined on a neighborhood of the closed ball119861(0 119903) Then

1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119872(119903 119892) 119903

minusm (10)

where119872(119903 119892) = max119911=119903

119892(119911) is themaximummodulus of119892(119911) (see [5] pp 1260)

Definition 3 Let 119892 R119899+1 rarr Cl0119899

be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then for 119903 gt 0 the maximum term of this special monogenicfunction is given by (see [5] pp 1260)

120583 (119903) = 120583 (119903 119892) = max|m|ge0

1003817100381710038171003817119886m1003817100381710038171003817119896m119903

m (11)

Also the index m with maximal length |m| for which maxi-mum term is achieved is called the central index and isdenoted by (see [5] pp 1260)

] (119903) = ] (119903 119891) = m (12)


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then the order 120588 and lower order 120582 of 119892(119911) are defined as (see[5] pp 1263)

120588

120582= lim119903rarrinfin

supinf

log log119872(119903 119892)

log 119903 (13)


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then the type 120590 and lower type120596 of special monogenic func-tion 119892(119911) having nonzero finite generalized order are definedas (see [5] pp 1270)

120590


supinf

log119872(119903 119892)

119903120588

(14)

For generalization of the classical characterizations of growthof entire functions Seremeta [6] introduced the concept ofthe generalized order and generalized type with the help ofgeneral growth functions as follows

Let 1198710 denote the class of functions ℎ(119909) satisfying thefollowing conditions

(i) ℎ(119909) is defined on [119886infin) and is positive strictlyincreasing and differentiable and it tends to infin as119909 rarr infin

(ii) lim119909rarrinfin

(ℎ[1 + 1120595(119909)119909]ℎ(119909)) = 1for every function120595(119909) such that120595(119909) rarr infin as 119909 rarrinfin

Let Λ denote the class of functions ℎ(119909) satisfying condi-tions (i) and

(i) lim119909rarrinfin

(ℎ(119888119909)ℎ(119909)) = 1(ii) for every 119888 gt 0 that is ℎ(119909) is slowly increasing

Following Srivastava and Kumar [7] and Kumar and Bala([8ndash10]) here we give definitions of generalized order gen-eralized lower order generalized type and generalized lowertype of special monogenic functions For special monogenicfunction 119892(119911) and functions 120572(119909) isin Λ 120573(119909) isin 1198710 we definethe generalized order 120588(120572 120573 119892) and generalized lower order120582(120572 120573 119892) of 119892(119911) as

120588 (120572 120573 119892)

120582 (120572 120573 119892)= lim119903rarrinfin

supinf

120572 [log119872(119903 119892)]

120573 (log 119903) (15)

If in above equation we put 120572(119903) = log 119903 and 120573(119903) = 119903 then weget definitions of order and lower order as defined byAbul-Ezand De Almeida (see [5] pp 1263) Hence their definitionsof order and lower order are special cases of our definitions

Further for 120572(119909) 120573minus1(119909) 120574(119909) isin 1198710 we define the gener-alized type 120590(120572 120573 120588 119892) and generalized lower type of specialmonogenic function 119892(119911) having nonzero finite generalizedorder as

120590 (120572 120573 120588 119892)

120596 (120572 120573 120588 119892)= lim119903rarrinfin

supinf

120572 [log119872(119903 119892)]

120573 [120574 (119903)120588

]

(16)

Journal of Complex Analysis 3

If in above equation we put 120572(119903) = 119903 120573(119903) = 119903 and 120574(119903) = 119903then we get definitions of type and lower type as defined byAbul-Ez and De Almeida (see [5] pp 1270) Hence theirdefinitions of type and lower type are special cases of ourdefinitions

Abul-Ez and De Almeida [5] have obtained the char-acterizations of order lower order type and lower typeof special monogenic functions in terms of their Taylorrsquosseries coefficients In the present paper we have obtainedthe characterizations of generalized order generalized lowerorder generalized type and generalized lower type of specialmonogenic functions in terms of their Taylorrsquos series coeffi-cients The results obtained by Abul-Ez and De Almeida [5]are special cases of our results

3 Main Results

We now prove the following

Theorem 6 Let 119892 R119899+1 rarr 1198621198970119899

be a special monogenicfunction whose Taylorrsquos series representation is given by (9) If120572(119909) isin Λ and 120573(119909) isin 1198710 then the generalized order 120588 of 119892(119911)is given as

120588 = 120588 (120572 120573 119892) = lim|m|rarrinfin

sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(17)

Proof Write

120579 = lim|m|rarrinfin

sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(18)

Now first we prove that 120588 ge 120579 The coefficients of amonogenic Taylorrsquos series satisfy Cauchyrsquos inequality that is

1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119872(119903 119892) 119903

minus|m| (19)

Also from (15) for arbitrary 120576 gt 0 and all 119903 gt 1199030(120576) we have

119872(119903 119892) le exp [120572minus1 120588120573 (log 119903)] 120588 = 120588 + 120576 (20)

Now from inequality (19) we get

1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119903minus|m| exp [120572minus1 120588120573 (log 119903)] (21)

Since (1radic119896m) le 1 (see [11] pp 148) so the above inequalityreduces to

1003817100381710038171003817119888m1003817100381710038171003817le 119903minus|m| exp [120572minus1 120588120573 (log 119903)] (22)

Putting 119903 = exp[120573minus1120572(|m|)120588] in the above inequality weget for all large values of |m|

1003817100381710038171003817119888m1003817100381710038171003817le exp [|m| minus |m| 120573minus1 120572 (|m|)

120588

] (23)

or

120573minus1

120572 (|m|) 120588 le 1 minus 1

|m|log 100381710038171003817

1003817119888m1003817100381710038171003817 (24)

or120572 (|m|)

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

le 120588 (25)

or

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

120573 log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

(26)

Since 120573(119909) isin 1198710 120573(1+119909) ≃ 120573(119909) Hence proceeding to limitsas |m| rarr infin we get


sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588 (27)

Since 120576 gt 0 is arbitrarily small so finally we get

120579 le 120588 (28)

Now we will prove that 120579 ge 120588 If 120579 = infin then there is nothingto prove So let us assume that 0 le 120579 lt infin Therefore for agiven 120576 gt 0 there exists 119899

0isin N such that for all multi-indices

m with |m| gt 1198990 we have

0 le

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

lt 120579 + 120576 = 120579 (29)

or

1003817100381710038171003817119888m1003817100381710038171003817le exp [minus |m| 120573minus1 120572 (|m|)

120579

] (30)

Now from the property of maximum modulus (see [11] pp148) we have

119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (31)

or

119872(119903 119892) le

infin

sum

|m|=0119896m119903|m| exp [minus |m| 120573minus1 120572 (|m|)

120579

] (32)

On the lines of the proof of the theorem given by Srivastavaand Kumar (see [7] Theorem 21 pp 666) we get

120588 le 120579 (33)

Combining this with inequality (28) we get (17) HenceTheorem 6 is proved

Next we prove the following


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Also let 120572(119909) 120573(119909) 120574(119909) isin 119871

0 and 0 lt 120588 lt infin then thegeneralized type 120590 of 119892(119911) is given as

120590 = 120590 (120572 120573 120588 119892) = lim|m|rarrinfin

sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(34)


Proof Write


sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(35)

Now first we prove that 120590 ge 120578 From (16) for arbitrary 120576 gt 0and all 119903 gt 119903

0(120576) we have

119872(119903 119892) le exp [120572minus1 120590120573 [120574(119903)120588]] (36)

where 120590 = 120590 + 120576 Now using (19) we get

1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119903minus|m| exp [120572minus1 120590120573 [120574 (119903)120588]] (37)

Now as in the proof of Theorem 6 here this inequalityreduces to

1003817100381710038171003817119888m1003817100381710038171003817le 119903minus|m| exp [120572minus1 120590120573 [120574 (119903)120588]] (38)

Putting 119903 = 120574minus1([120573minus1(1120590)120572(|m|120588)]1120588) we get

1003817100381710038171003817119888m1003817100381710038171003817le 120574minus1

([120573minus1

1

120590

120572(

|m|120588

)]

1120588

)

minus|m|

times exp(|m|120588

)

(39)

or

1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|ge 120574minus1

([120573minus1

1

120590

120572(

|m|120588

)]

1120588

) exp(minus1120588

)

(40)

or

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120590 (41)

Now proceeding to limits as |m| rarr infin we get

120578 le 120590 (42)


120578 le 120590 (43)




0 le

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120578 + 120576 = 120578 (44)

or

1003817100381710038171003817119888m1003817100381710038171003817le (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(45)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (46)

or

119872(119903 119892) le

infin

sum

|m|=0119896m119903|m|

times (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(47)


120590 le 120578 (48)

Combining this with (43) we get (34) Hence Theorem 7 isproved

Next we have the following


be a special monogenicfunction whose Taylorrsquos series representation is given by (9) If120572(119909) isin Λ and 120573(119909) isin 1198710 then the generalized lower order 120582 of119892(119911) satisfies

120582 = 120582 (120572 120573 119892) ge lim|m|rarrinfin

inf 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(49)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (50)

is a nondecreasing function of 119905 then equality holds in (49)

Proof The proof of the above theorem follows on the lines ofthe proof ofTheorem 6 and [7]Theorem 24 (pp 674) Hencewe omit the proof



be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Also let (119909) 120573(119909) 120574(119909) isin 119871

0 and 0 lt 120588 lt infin then thegeneralized lower type 120596 of 119892(119911) satisfies

120596 = 120596 (120572 120573 120588 119892) ge lim|m|rarrinfin

inf120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(51)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (52)




Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author is very thankful to the referee for the valuablecomments and observations which helped in improving thepaper

References

[1] R de Almeida and R S Krauszlighar ldquoOn the asymptotic growthof entire monogenic functionsrdquo Zeitschrift fur Analysis und ihreAnwendungen vol 24 no 4 pp 791ndash813 2005

[2] D Constales R de Almeida and R S Krausshar ldquoOn thegrowth type of entire monogenic functionsrdquo Archiv der Mathe-matik vol 88 no 2 pp 153ndash163 2007

[3] D Constales R de Almeida and R S Krausshar ldquoOn therelation between the growth and the Taylor coefficients of entiresolutions to the higher-dimensional Cauchy-Riemann systeminR119899+1rdquo Journal of Mathematical Analysis and Applications vol327 no 2 pp 763ndash775 2007

[4] M A Abul-Ez and D Constales ldquoBasic sets of polynomials inClifford analysisrdquo Complex Variables Theory and Applicationvol 14 no 1ndash4 pp 177ndash185 1990

[5] MAAbul-Ez andRDeAlmeida ldquoOn the lower order and typeof entire axially monogenic functionsrdquo Results in Mathematicsvol 63 no 3-4 pp 1257ndash1275 2013

[6] M N Seremeta ldquoOn the connection between the growth ofthe maximum modulus of an entire function and the moduliof the coefficients of its power series expansionrdquoThe AmericanMathematical Society Translations vol 88 no 2 pp 291ndash3011970

[7] G S Srivastava and S Kumar ldquoOn the generalized order andgeneralized type of entire monogenic functionsrdquo Demon Mathvol 46 no 4 pp 663ndash677 2013

[8] S Kumar and K Bala ldquoGeneralized type of entire monogenicfunctions of slow growthrdquo Transylvanian Journal of Mathemat-ics and Mechanics vol 3 no 2 pp 95ndash102 2011

[9] S Kumar and K Bala ldquoGeneralized order of entire monogenicfunctions of slow growthrdquo Journal of Nonlinear Science and itsApplications vol 5 no 6 pp 418ndash425 2012

[10] S Kumar and K Bala ldquoGeneralized growth of monogenicTaylor series of finite convergence radiusrdquoAnnali dellrsquoUniversitadi Ferrara VII Scienze Matematiche vol 59 no 1 pp 127ndash1402013

[11] M A Abul-Ez and D Constales ldquoLinear substitution forbasic sets of polynomials in Clifford analysisrdquo PortugaliaeMathematica vol 48 no 2 pp 143ndash154 1991

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


If 119880 sube R119899+1 is an open set then a function 119892 119880 rarr Cl0119899is

called left (right) monogenic at a point 119911 isin 119880 if 119863119892(119911) = 0(119892119863(119911) = 0) The functions which are left (right) monogenicin the whole space are called left (right) entire monogenicfunctions

Following Abul-Ez and Constales [4] we consider theclass of monogenic polynomials 119901m of degree |m| defined as

119901m (119911) =infin

sum

119894+119895=|m|

((119899 minus 1) 2)119894

119894

((119899 + 1) 2)119895

119895

(119911)119894

119911119895

(6)

Let 119908119899be 119899-dimensional surface area of 119899 + 1-dimensional

unit ball and let 119878119899 be 119899-dimensional sphere Then the classof monogenic polynomials described in (6) satisfies (see [5]pp 1259)

1

119908mint

119878119899

119901m (119911)119901l (119911) 119889119878119911 = 119896m120575|m||l| (7)

Also following Abul-Ez and De Almeida [5] we have

max119911=119903

1003817100381710038171003817119901m (119911)

1003817100381710038171003817= 119896m119903

m (8)

2 Preliminaries

Now following Abul-Ez and De Almeida [5] we give somedefinitions which will be used in the next section

Definition 1 Let Ω be a connected open subset of R119899+1

containing the origin and let 119892(119911) be monogenic in Ω Then119892(119911) is called specialmonogenic inΩ if and only if its Taylorrsquosseries near zero has the form (see [5] pp 1259)

119892 (119911) =

infin

sum

|m|=0119901m (119911) 119888m 119888m isin Cl0119899 (9)

Definition 2 Let 119892(119911) = suminfin|m|=0 119901m(119911)119888m be a special mono-

genic function defined on a neighborhood of the closed ball119861(0 119903) Then

1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119872(119903 119892) 119903

minusm (10)

where119872(119903 119892) = max119911=119903

119892(119911) is themaximummodulus of119892(119911) (see [5] pp 1260)


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then for 119903 gt 0 the maximum term of this special monogenicfunction is given by (see [5] pp 1260)

120583 (119903) = 120583 (119903 119892) = max|m|ge0

1003817100381710038171003817119886m1003817100381710038171003817119896m119903

m (11)

Also the index m with maximal length |m| for which maxi-mum term is achieved is called the central index and isdenoted by (see [5] pp 1260)

] (119903) = ] (119903 119891) = m (12)


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then the order 120588 and lower order 120582 of 119892(119911) are defined as (see[5] pp 1263)

120588


supinf

log log119872(119903 119892)

log 119903 (13)


be a special monogenicfunction whose Taylorrsquos series representation is given by (9)Then the type 120590 and lower type120596 of special monogenic func-tion 119892(119911) having nonzero finite generalized order are definedas (see [5] pp 1270)

120590


supinf

log119872(119903 119892)

119903120588

(14)

For generalization of the classical characterizations of growthof entire functions Seremeta [6] introduced the concept ofthe generalized order and generalized type with the help ofgeneral growth functions as follows

Let 1198710 denote the class of functions ℎ(119909) satisfying thefollowing conditions

(i) ℎ(119909) is defined on [119886infin) and is positive strictlyincreasing and differentiable and it tends to infin as119909 rarr infin

(ii) lim119909rarrinfin

(ℎ[1 + 1120595(119909)119909]ℎ(119909)) = 1for every function120595(119909) such that120595(119909) rarr infin as 119909 rarrinfin

Let Λ denote the class of functions ℎ(119909) satisfying condi-tions (i) and

(i) lim119909rarrinfin

(ℎ(119888119909)ℎ(119909)) = 1(ii) for every 119888 gt 0 that is ℎ(119909) is slowly increasing

Following Srivastava and Kumar [7] and Kumar and Bala([8ndash10]) here we give definitions of generalized order gen-eralized lower order generalized type and generalized lowertype of special monogenic functions For special monogenicfunction 119892(119911) and functions 120572(119909) isin Λ 120573(119909) isin 1198710 we definethe generalized order 120588(120572 120573 119892) and generalized lower order120582(120572 120573 119892) of 119892(119911) as

120588 (120572 120573 119892)

120582 (120572 120573 119892)= lim119903rarrinfin

supinf

120572 [log119872(119903 119892)]

120573 (log 119903) (15)

If in above equation we put 120572(119903) = log 119903 and 120573(119903) = 119903 then weget definitions of order and lower order as defined byAbul-Ezand De Almeida (see [5] pp 1263) Hence their definitionsof order and lower order are special cases of our definitions

Further for 120572(119909) 120573minus1(119909) 120574(119909) isin 1198710 we define the gener-alized type 120590(120572 120573 120588 119892) and generalized lower type of specialmonogenic function 119892(119911) having nonzero finite generalizedorder as

120590 (120572 120573 120588 119892)

120596 (120572 120573 120588 119892)= lim119903rarrinfin

supinf

120572 [log119872(119903 119892)]

120573 [120574 (119903)120588

]

(16)




3 Main Results




120588 = 120588 (120572 120573 119892) = lim|m|rarrinfin

sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(17)

Proof Write


sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(18)


1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119872(119903 119892) 119903

minus|m| (19)


119872(119903 119892) le exp [120572minus1 120588120573 (log 119903)] 120588 = 120588 + 120576 (20)


1003817100381710038171003817119888m1003817100381710038171003817le

1






120588

] (23)

or

120573minus1

120572 (|m|) 120588 le 1 minus 1

|m|log 100381710038171003817

1003817119888m1003817100381710038171003817 (24)

or120572 (|m|)

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

le 120588 (25)

or

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

120573 log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

(26)



sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588 (27)


120579 le 120588 (28)




0 le

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

lt 120579 + 120576 = 120579 (29)

or


120579

] (30)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (31)

or

119872(119903 119892) le

infin

sum


120579

] (32)


120588 le 120579 (33)






120590 = 120590 (120572 120573 120588 119892) = lim|m|rarrinfin

sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(34)


Proof Write


sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(35)


0(120576) we have

119872(119903 119892) le exp [120572minus1 120590120573 [120574(119903)120588]] (36)


1003817100381710038171003817119888m1003817100381710038171003817le

1





1003817100381710038171003817119888m1003817100381710038171003817le 120574minus1

([120573minus1

1

120590

120572(

|m|120588

)]

1120588

)

minus|m|

times exp(|m|120588

)

(39)

or

1003817100381710038171003817119888m1003817100381710038171003817


([120573minus1

1

120590

120572(

|m|120588

)]

1120588

) exp(minus1120588

)

(40)

or

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120590 (41)


120578 le 120590 (42)


120578 le 120590 (43)




0 le

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120578 + 120576 = 120578 (44)

or

1003817100381710038171003817119888m1003817100381710038171003817le (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(45)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (46)

or

119872(119903 119892) le

infin

sum

|m|=0119896m119903|m|

times (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(47)


120590 le 120578 (48)






inf 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(49)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (50)








inf120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(51)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (52)






Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3 Main Results




120588 = 120588 (120572 120573 119892) = lim|m|rarrinfin

sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(17)

Proof Write


sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(18)


1003817100381710038171003817119888m1003817100381710038171003817le

1

radic119896m119872(119903 119892) 119903

minus|m| (19)


119872(119903 119892) le exp [120572minus1 120588120573 (log 119903)] 120588 = 120588 + 120576 (20)


1003817100381710038171003817119888m1003817100381710038171003817le

1






120588

] (23)

or

120573minus1

120572 (|m|) 120588 le 1 minus 1

|m|log 100381710038171003817

1003817119888m1003817100381710038171003817 (24)

or120572 (|m|)

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

le 120588 (25)

or

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588

120573 1 + log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

120573 log 1003817100381710038171003817119888m1003817100381710038171003817

minus1|m|

(26)



sup 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

le 120588 (27)


120579 le 120588 (28)




0 le

120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

lt 120579 + 120576 = 120579 (29)

or


120579

] (30)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (31)

or

119872(119903 119892) le

infin

sum


120579

] (32)


120588 le 120579 (33)






120590 = 120590 (120572 120573 120588 119892) = lim|m|rarrinfin

sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(34)


Proof Write


sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(35)


0(120576) we have

119872(119903 119892) le exp [120572minus1 120590120573 [120574(119903)120588]] (36)


1003817100381710038171003817119888m1003817100381710038171003817le

1





1003817100381710038171003817119888m1003817100381710038171003817le 120574minus1

([120573minus1

1

120590

120572(

|m|120588

)]

1120588

)

minus|m|

times exp(|m|120588

)

(39)

or

1003817100381710038171003817119888m1003817100381710038171003817


([120573minus1

1

120590

120572(

|m|120588

)]

1120588

) exp(minus1120588

)

(40)

or

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120590 (41)


120578 le 120590 (42)


120578 le 120590 (43)




0 le

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120578 + 120576 = 120578 (44)

or

1003817100381710038171003817119888m1003817100381710038171003817le (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(45)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (46)

or

119872(119903 119892) le

infin

sum

|m|=0119896m119903|m|

times (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(47)


120590 le 120578 (48)






inf 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(49)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (50)








inf120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(51)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (52)






Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Proof Write


sup120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(35)


0(120576) we have

119872(119903 119892) le exp [120572minus1 120590120573 [120574(119903)120588]] (36)


1003817100381710038171003817119888m1003817100381710038171003817le

1





1003817100381710038171003817119888m1003817100381710038171003817le 120574minus1

([120573minus1

1

120590

120572(

|m|120588

)]

1120588

)

minus|m|

times exp(|m|120588

)

(39)

or

1003817100381710038171003817119888m1003817100381710038171003817


([120573minus1

1

120590

120572(

|m|120588

)]

1120588

) exp(minus1120588

)

(40)

or

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120590 (41)


120578 le 120590 (42)


120578 le 120590 (43)




0 le

120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

le 120578 + 120576 = 120578 (44)

or

1003817100381710038171003817119888m1003817100381710038171003817le (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(45)


119872(119903 119892) le

infin

sum

|m|=0

1003817100381710038171003817119888m1003817100381710038171003817119896m119903|m| (46)

or

119872(119903 119892) le

infin

sum

|m|=0119896m119903|m|

times (120574minus1

[120573minus1

1

120578

120572(

|m|120588

)]

1120588

)

minus|m|

119890|m|120588

(47)


120590 le 120578 (48)






inf 120572 (|m|)120573 log 100381710038171003817

1003817119888m1003817100381710038171003817

minus1|m|

(49)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (50)








inf120572 (|m| 120588)

120573 [120574 (11989011205881003817100381710038171003817119888m1003817100381710038171003817

minus1|m|)

120588

]

(51)

Further if

120595 (119905) = max|m|=119905

1003817100381710038171003817119888m1003817100381710038171003817119896m

1003817100381710038171003817119888m10158401003817100381710038171003817119896m1015840

10038161003816100381610038161003816m101584010038161003816100381610038161003816= |m| + 1 (52)






Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgment


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Generalized Growth of Special Monogenic ...downloads.hindawi.com/archive/2014/510232.pdf · Research Article Generalized Growth of Special Monogenic Functions SusheelKumar