Convolution operators in the geometric function theory

Shareef Zahid, Hussain Saqib, Darus, Darus Maslina

Description

The study of operators plays a vital role in mathematics. To define an operator using the convolution theory, and then study its properties, is one of the hot areas of current ongoing research in the geometric function theory and its related fields. In this survey-type article, we discuss historic development and exploit the strengths and properties of some differential and integral convolution operators introduced and studied in the geometric function theory. It is hoped that this article will be beneficial for the graduate students and researchers who intend to start work in this field. MSC: 30C45, 30C50.

Sujets

• Analytic function
• Convolution
• Hypergeometric function
• Differential operator
• Integral transform

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 26
• Langue: English

Extrait

Shareefetal.JournalofInequalitiesandApplications2012,2012:213
http://www.journalofinequalitiesandapplications.com/content/2012/1/213
RESEARCH OpenAccess
Convolutionoperatorsinthegeometric
functiontheory
1 2 1*ZahidShareef ,SaqibHussain andMaslinaDarus
*Correspondence:[e-mail]
1SchoolofMathematicalSciences, Abstract
FacultyofScienceandTechnology,
Thestudyofoperatorsplaysavitalroleinmathematics.TodefineanoperatorusingUniversitiKebangsaanMalaysia,
Bangi,Selangor43600,Malaysia theconvolutiontheory,andthenstudyitsproperties,isoneofthehotareasofcurrent
Fulllistofauthorinformationis ongoingresearchinthegeometricfunctiontheoryanditsrelatedfields.Inthis
availableattheendofthearticle
survey-typearticle,wediscusshistoricdevelopmentandexploitthestrengthsand
propertiesofsomedifferentialandintegralconvolutionoperatorsintroducedand
studiedinthegeometricfunctiontheory.Itishopedthatthisarticlewillbebeneficial
forthegraduatestudentsandresearcherswhointendtostartworkinthisfield.
MSC: 30C45;30C50
Keywords: analyticfunctions;convolution;hypergeometricfunction;differential
operator;integraloperator
1 Introduction
LetAdenotetheclassoffunctionsoftheform
∞
nf(z)=z+ a z , (.)n
n=
whichareanalyticintheopenunitdiscE= {z: |z|<},centeredatoriginandnormalized
bytheconditionsf()=andf ()=.Also,letS ⊂Abetheclassoffunctionswhichare
univalentinE.TheconvolutionorHadamardproductoftwofunctionsf,g ∈Aisdenoted
byf ∗g andisdefinedas
∞
n(f ∗g)(z)=z+ a b z,(.)n n
n=
∞ nwhere f(z)isgivenby(.), and g(z)= z + b z . Note that the convolution of twonn=
functionsisagainanalyticinE,i.e.f ∗g)(z) ∈A.
For the complex parameters a, b and c with c
=,–,–,–,...,theGausshypergeometricfunction F (a,b,c;z)isdefinedas 
∞(a) (b)n n nF (a,b,c;z)= z 
(c) n!nn=
∞ (a) (b)n– n– n–=+ z,(.)
(c) (n–)!n–n=
© 2012 Shareef et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
inanymedium,providedtheoriginalworkisproperlycited.Shareefetal.JournalofInequalitiesandApplications2012,2012:213 Page2of11
http://www.journalofinequalitiesandapplications.com/content/2012/1/213
whereandinthesubscriptofF
merelysignifythenumberofparametersinthenumerantorandthedenominatorofthecoefficientofz respectively.Also,(α) isthePochhammern
symbol(ortheshiftedfactorial)definedas
⎧
⎨, ifn=,
(α) = (.)n
⎩α(α+)(α+)···(α+n–), ifn
=.
ThePochhammersymbolisrelatedtothefactorialandthegammafunctionsbytherelation
(α+n)
(α) = .n
(α)
∗By S , C and K,wemeanthesubclassesof S consisting of starlike, with respect to the
origin,convexandclose-to-convexunivalentfunctionsinE respectively.
∗TheseclassesofS andC arerelatedtoeachotherbytheAlexanderrelation[,].Later
Libera [] introduced an integral operator and showed that these two classes are closed
under this operator. Bernardi [] gave a generalized operator and studied its
properties.
Ruscheweyh[],NoorandNoor[,],Noor[]andmanyothers,forexample,[–],defined new operators and studied various classes of analytic and univalent functions
generalizing a number of previously known classes and at times discovering new classes of
analyticfunctions.
2 Convolutionoperators
The study of operators plays an important role in the geometric function theory. Many
differential and integral operators can be written in terms of convolution of certain
analytic functions. It is observed that this formalism brings an ease in further
mathematical
explorationandalsohelpstounderstandthegeometricpropertiesofsuchoperatorsbetter.
The importance of convolution in the theory of operators may be understood by the
followingsetofexamplesgivenbyBarnardandKellogg[].
Letf ∈Aand :A →Aforeachi∈{,,,}bethelinearoperatorsdefinedasi
f(z)=zf (z) (Alexanderdifferential[]),
(zf(z)) f(z)+zf (z)
f(z)= = (Livingston[]),
 
z f(ζ)
f(z)= dζ (Alexanderintegral[,]),
ζ
z
f(z)= f(ζ)dζ (Libera[]).
z 
Note that the first two of the above operators are differential and others are integral in
nature.Now,eachoftheseoperatorscanbewrittenasaconvolutionoperatorasfollows:
f(z)=(h ∗f)(z), i∈{,,,},i iShareefetal.JournalofInequalitiesandApplications2012,2012:213 Page3of11
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where
∞ ∞ z z n+ z–n n h (z)=z+ nz = , h (z)=z+ z = ,  (–z)  (–z)
n= n=
∞ ∞   –[z+log(–z)]n n
h (z)=z+ z =–log(–z), h (z)=z+ z = . 
n n+ z
n= n=
This shows how differential and integral operators may be written in terms of
convolutionoffunctions.Also,notethatonceweputtheseoperatorsintoconvolutionformalism,
it becomes easy to conclude that the Alexander differential operator is the inverse of the
Alexander integral operator, whereas the Livingston operator is the inverse of the
Libera operator. For further discussion on the importance of the convolution operation, we
recommendthereadertogothroughtheclassicalworkofRuscheweyh[].
Now, we give a brief survey of some convolution operators studied in the geometric
functiontheoryinchronologicalorderandmentionsomerelatedworks.
2.1 ThegeneralizedBernardioperator(1969)
ConsidertheoperatorJ :A →Agivenbyη
z+ η η–J f(z)= t f(t)dt, Re(η)>–. (.)η ηz 
NotethattheAlexanderintegralandLiberaoperatorsarespecialcasesofJ for η=andη
η=respectively.Now,J f(z)canequivalentlybeputintoaconvolutionformalismasη
J f(z)=z F (,+ η,+ η;z) ∗f(z), (.)η  
where F (, + η,+ η;z) is the Gaussian hypergeometric function given by (.). The 
∗operator J was introduced by Bernardi []. In [], it was also shown that the classes Sη
and C are closed under this operator, i.e., the generalized Bernardi operator maps the
∗ ∗classesof S and C ontotheclassesof S and C respectively.Someofotherworksonthe
Bernardioperatorinclude[]and[]andreferencestherein.
2.2 Ruscheweyhderivativeoperator(1975)
λUsingthetechniqueofconvolution,Ruscheweyh[]definedtheoperatorD ontheclass
ofanalyticfunctionsAas
zλD f(z)= ∗f(z), λ ∈R,λ>–. (.)
λ+(–z)
For λ=m ∈N =N∪{},weobtain
m– (m)z(z f(z))mD f(z)=.(.)
m!
mThe expression D f(z) is called an mth-order Ruscheweyh derivative of f(z). Note that
  D f(z)=f(z)whichisidentityoperator,andD f(z)=zf (z)= ,theAlexanderdifferentialShareefetal.JournalofInequalitiesandApplications2012,2012:213 Page4of11
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operator.Itcanalsobeshownthatthisoperatorishypergeometricinnatureas
λD f(z)=z F (λ+,,;z) ∗f(z). (.) 
λThefollowingidentityiseasilyestablishedfortheoperatorD :
λ λ+ λz D f =(λ+)D f – λD f.
Manyauthors,see,forexample,[–],haveusedtheRuscheweyhoperatortodefineand
investigatethepropertiesofcertainknownandnewclassesofanalyticfunctions.
2.3 Carlson-Shafferoperator(1984)
Carlson and Shaffer []usedtheHadamardproducttodefinealinearoperator L(a,c):
A →Aby
L(a,c)f(z)= ϕ(a,c;z) ∗f(z), (.)
where
∞ ∞ (a) (a)n n–n+ nϕ(a,c;z)= z =z+ z , c =,–,–,...
(c) (c)n n–n= n=
∞ (c)(a+n–) n=z+ z,(.)
(a)(c+n–)
n=
is the incomplete beta function with ϕ(a,c;z) ∈A.Using(.)and(.), we can establish
arelationbetweenhypergeometric