On Meromorphic Harmonic Functions with Respect to -Symmetric Points

K Al-Shaqsi,M Darus

doi:10.1155/2008/259205

Abstract

In our previous work in this journal in 2008, we introduced the generalized derivative operator for . In this paper, we introduce a class of meromorphic harmonic function with respect to -symmetric points defined by . Coefficient bounds, distortion theorems, extreme points, convolution conditions, and convex combinations for the functions belonging to this class are obtained.

Highlights

A continuous function f u iv is a complex valued harmonic function in a domain D ⊂ C if both u and v are real harmonic in D
A necessary and sufficient condition for f to be locally univalent and orientation preserving in D is that |h | > |g | in D see 1
In 3, the authors introduced the operator Djm for f ∈ SH which is the class of functions f h g that are harmonic univalent and sense-preserving in the unit disk

Summary

Introduction

A continuous function f u iv is a complex valued harmonic function in a domain D ⊂ C if both u and v are real harmonic in D. Hengartner and Schober 2 investigated functions harmonic in the exterior of the unit disk U {z : |z| > 1} They showed that complex valued, harmonic, sense preserving, univalent mapping f must admit the representation fzhzgzA log |z|, 1.1 where h z and g z are defined by. In 3 , the authors introduced the operator Djm for f ∈ SH which is the class of functions f h g that are harmonic univalent and sense-preserving in the unit disk. M ∈ N0, 0 ≤ α < 1 and k ≥ 1, let MHSsk j, m, α denote the class of meromorphic harmonic functions f of the form 1.3 such that. We will give a sufficient condition for functions f h g, where h and g given by 1.3 to be in the class MHSsk j, m, α.

Coefficient bounds

Distortion bounds and extreme points

Convolution and convex combination