Subclass of Multivalent Harmonic Functions Defi ned by Wright Generalized Hypergeometric Functions

M K Aouf,E A Adwan,A O Moustafa

doi:10.1155/2013/397428

Abstract

We introduce a new class of multivalent harmonic functions defi ned by Wright generalized hypergeometric function. Coefficient estimates, extreme points, distortion bounds, and convex combination for functions belonging to this class are obtained.

Highlights

A continuous complex-valued function f = u + iV defined in a connected complex domain D is said to be harmonic in D if both u and V are real harmonic in D
Ahuja and Jahangiri [2] defined the class Hp(p ∈ N = {1, 2, . . .}) consisting of all p-valent harmonic functions f = h + g that are sense-preserving in U and h, g are of the form
Let Hp([α1, A1, B1], q, s; γ) be the subclass of Hp([α1, A1, B1], q, s; γ) consisting of functions f = h + g such that h and g are of the form

Summary

Introduction

Denote by SH the class of functions f of the form (1) that are harmonic univalent and sense preserving in the unit disc U = {z : |z| < 1} for which f(0) = fz(0) − 1 = 0. By using the generalized hypergeometric function, Dziok and Srivastava [5] introduced a linear operator. Dziok and Raina in [6] and Aouf and Dziok in [7] extended this linear operator by using Wright generalized hypergeometric function. Let Hp([α1, A1, B1], q, s; γ) be the subclass of Hp([α1, A1, B1], q, s; γ) consisting of functions f = h + g such that h and g are of the form h (z) = zp − ∑ anzn, n=1+p g (z) = ∑ bnzn, n=p.

Coefficient Estimates

Distortion Theorem

Extreme Points