.lournal oflvluthematical und.Phl,siculScience:li,Ittl.-1(). Ntt.6.De.cemhu 199(t Printed in India
3A7
On Starlike Functionswith Respectto N-Ply Conjugateand SymmetricConjugatePoints M. S. KASI Department of lvlathernatis, l,oyala College, Madras, INDIA
V. RAVICHANDRAN Deparlrnent of Mathenntils, Anna lhrivenity, Madra-s-600025, INI)IA.
Abstract The classof starlikefunclionswith respectto n-ply corrjugate and syrunetriccorrjugatepoirrlsare of functionsarealsocon^sidered. introducedandstudied.Sorrrerelatedclasses
l. Introduction Let
A denotethe class of all analytic fuuctions "f (z) defined on the unit disk
ry =lz,lzl< l) with the normalizations "f(O)=0,/'(0)= l.
Let 5' denote the classof all
in A. Let .S*(a),((a) and (:(a) denotethe usual classes of starlikc. univalentfunctions convexand closeto convexfunctionsof order a 0 < q, < I respectivelydefinedby
.5'*(a)= lf el; Re #
> u,z e(-il
K(a)= lf el; Re{l.4r-l!_.', > u.z e(J} l'(z)
Cl(c,)= Lf e"4;Re19> o, for sorneg € K. z eLl\ g'(z)
M. S. KASI and v. RAVICIIANDRAN
A function .f e I is starlikewith respectto symrnetricpointsin [/ if
,l'' tr) x"[ ], o.z e(r. (z).f(-z)) lf (4). A functiolrf e A is starlikcwith rcspect This classis introducedandstudiedby Sakagrrchi to conjugatepointsin t/ is / satisfiesthe condition
n " { . ' f '-.f (2= }tn, z eu. lf (z) (z)) A function ./ e I is surlike with respectto symmetricconjugatepointsin ( I if it sntishcs I zf'tz\ Re{4'
I -}>0' z e(!'
l.f (z)- .f(-z)J
Denotetheclasses consisting of thesefunctions Thcscclasscs UVSi and5l respectively. are introducedby El-Ashwahand Thomas(l). Thesefunctionsare close lo convex. Sokol(fi) introduced two niore pararrreter in this classand obtainedstructural fornrurla,tlrc coclliciorl estirnate,the radiusof convexityand resultsaboutthe neighbourhoods of functions. Slrannrugarrr and Ravichandran(5) also studied the classof starlikefunctionswith respectlo syrnnrctric points.
In this paperwe introducethe classof functionsstarlikewith respectto n -ply conjugate points and certain relatedclassesof functionsand obtain structuralfomrula, distortion theorcnr and convolutiontheorems.
2. IDefinitionsand Preliminaries
Inthefollowing definitio\rs, n > l,wn = I and 0
lcl
On Starlike Functions with Respect to N-Ply Conjugate ...
convex and close to convex functions of order a 0 < q, < I respectively ... inition 2.3 : A function f e A is close to convex rvith respect to n -ply cortjugatc poittts of.