Applied Mathematics Letters Some Convolution Conditions for ... - Core

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Applied Mathematics Letters

Applied Mathematics Letters 16 (2003) 13-16

www.elsevier.com/locate/aml

S o m e Convolution C o n d i t i o n s for Starlikeness and C o n v e x i t y of M e r o m o r p h i c a l l y Multivalent Functions JIN-LIN LIU Department of Mathematics,YangzhouUniversity Yangzhou 225002, Jiangsu, P.R. China j lliucn@yahoo, com. cn H. M . SItIVASTAVA Department of Mathematics and Statistics,Universityof Victoria Victoria, BritishColumbia V 8 W 3P4, Canada harimsri~nath, uvic. ca

(Received December 2001; accepted February 2002) Abstract--By making use of the Hadamard product (or convolution), the authors derive necessary and sufficient conditions for a suitably normalized meromorphically p-valent function to be in the familiar classes of meromorphic p-valently starlike and meromorphic p-valently convex functions in the punctured unit disk. ~ 2002 Elsevier Science Ltd. All rights reserved. Keywords--Hadamard product (or convolution), Meromorphic functions, Multivalent functions, Analytic functions, Starlike functions, Convex functions.

1. I N T R O D U C T I O N

AND

DEFINITIONS

Let ~-~p denote the class of functions f normalized by c~

f (Z) = z -p + E a k

z k-p

(p e N := {1,2,3,...}),

(1.1)

k=l

which are analytic and p-valent in the punctured unit disk U* := {z: z e C and 0 < [z I < 1} = U\{O}. A function f E ~-~p is said to be in the class S~ (a) of meromorphic p-valently starlike functions of order a in U* if and only if Yt\ f(z) ]