On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator

Abdeljabbar Talal Yousef,Zabidin Salleh

doi:10.3390/axioms9010032

Abstract

In this paper, a subclass of complex-valued harmonic univalent functions defined by a generalized linear operator is introduced. Some interesting results such as coefficient bounds, compactness, and other properties of this class are obtained.

Highlights

Let H represent the continuous harmonic functions which are harmonic in the open unit disk
A harmonic function in U could be expressed as f = h + g, where h and g are in A, h is the analytic part of f, g is the co-analytic part of f and h0 (z) > g0 (z) is a necessary and sufficient condition for f to be locally univalent and sense-preserving in U
Let SH represents the functions of the form f = h + g which are harmonic and univalent in

Summary

Introduction

Let H represent the continuous harmonic functions which are harmonic in the open unit disk. U = {z : z ∈ C, |z| < 1} and let A be a subclass of H which represents the functions which are analytic in. Let SH represents the functions of the form f = h + g which are harmonic and univalent in. Many researchers have studied the class SH and even investigated some subclasses of it. The operator Iδ,μ,λ,ς,τ f (z) generalizes the following differential operators: If f ∈ A, when we take μ = 1, λ = 0, δ = 0, τ = 1, ς = 1 we obtain I0,τ, f (z) was δ,ς introduced and studied by Ramadan and Darus [6]. Denote by SH0 (δ, μ, λ, ς, τ, m, A, B) the subclass of SH0 consisting of functions of the form (1) that satisfy the condition m+1.

Coefficient Bounds

Compactness and Convex

Extreme Points