On the Generalization of a Class of Harmonic Univalent Functions Defined by Differential Operator

Aqeel Ketab Al-Khafaji,Salwa Salman Abed,Waggas Galib Atshan

doi:10.3390/math6120312

Aqeel Ketab Al-Khafaji, Salwa Salman Abed + Show 1 more

Open Access

https://doi.org/10.3390/math6120312

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Abstract

In this article, a new class of harmonic univalent functions, defined by the differential operator, is introduced. Some geometric properties, like, coefficient estimates, extreme points, convex combination and convolution (Hadamard product) are obtained.

Highlights

A continuous function f = u + iv is a complex-valued harmonic function in a complex domain C if both u and v are real harmonic
In any connected domain B ⊂ C, we can write f = h + g, where h and g are analytic in B
Denote by S H the family of harmonic functions f = h + g, which are univalent and sense-preserving in the open unit disc U = {z ∈ C : |z| < 1} where h and g are analytic in B and f is normalized by f (0) = h(0) = f z (0) − 1 = 0

Summary

Introduction

Clunie and Sheil-Small [1] observed that a necessary and sufficient condition for the harmonic functions f = h + g to be locally univalent and sense-preserving in B is that |h0 (z)| > | g0 (z)|, (z ∈ B). Denote by S H the family of harmonic functions f = h + g, which are univalent and sense-preserving in the open unit disc U = {z ∈ C : |z| < 1} where h and g are analytic in B and f is normalized by f (0) = h(0) = f z (0) − 1 = 0. In 1984 Clunie and Sheil-Small [1] investigated the class S H , as well as its geometric subclass and obtained some coefficient bounds. Coefficient conditions, distortion bounds, extreme points, convex combination and radii of convexity for this class are obtained

Main Results

Extreme Points

Convex Combination

Integral Operator