A New Subclass of Meromorphic Close-to-Convex Functions

Amit Soni,Shashi Kant

doi:10.1155/2013/629394

Abstract

A new subclass MK(t, A, B) of meromorphic close-to-convex functions, defined by means of subordination, is investigated. Some results such as inclusion relationship, coefficient inequality, convolution property, and distortion property for this class are derived. The results obtained here are extension of earlier known work.

Highlights

IntroductionWhich are analytic in the punctured open unit disk DD D DDD D zz zzzzz zzzz z zz. For any two analytic function ff and gg in UU UUU ⋃{0}, we say that ff is subordinate to gg in UU, written as ffffff f ffffff, if there exists a Schwarz function wwwwww such that fffffffffffffffff, for zz z zz
Let ∑ denote the class of functions ff of the form ff = 1 zz + ∞󵠈󵠈aannzznn, nnnn (1)which are analytic in the punctured open unit disk DD D DDD D zz zzzzz zzzz z zz
Let ∑ denote the class of functions ff of the form ff which are analytic in the punctured open unit disk DD D DDD D zz zzzzz zzzz z zz

Summary

Introduction

Which are analytic in the punctured open unit disk DD D DDD D zz zzzzz zzzz z zz. For any two analytic function ff and gg in UU UUU ⋃{0}, we say that ff is subordinate to gg in UU, written as ffffff f ffffff, if there exists a Schwarz function wwwwww such that fffffffffffffffff, for zz z zz. Wang et al [1] introduced and discussed the class MK of meromorphic functions ffffffff which satis es the inequality ff′(zz) gg(zz) gg(−zz). E class MMMM is very close to the interesting subclass KKss of close-to-convex function for analytic function introduced and studied by Gao and Zhou [2]. A function ffffffff is said be in the class ∑(AAAAAA if it satis es the following subordination relation: ff′(zz) ≺ 1 + AAAA ,. Instead of −zzzzzzzzzzzzzzz in (6) and (7), we can consider ttttttttttttttttt t t tttt tt, because if gggggg g gggg∗(1/2), tttttttttttttttt is a meromorphic starlike function, which motivates us to de ne a new subclass MMMMMMMMMMMMMM of meromorphic close-to-convex functions as follows. We aim at proving results such as inclusion relationship, coefficient bounds, and distortion theorem for the class MMMMMMMMMMMMMM

Properties of Meromorphic Starlike Functions

Coefficient Estimates

Distortion Theorem