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

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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
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