Harmonic Starlike Functions with Respect to Symmetric Points

Nak Eun Cho,Jacek Dziok

doi:10.3390/axioms9010003

Abstract

In the paper we define classes of harmonic starlike functions with respect to symmetric points and obtain some analytic conditions for these classes of functions. Some results connected to subordination properties, coefficient estimates, integral representation, and distortion theorems are also obtained.

Highlights

We denote by H the class of complex-valued harmonic functions in the unit disc U := {z : |z| < r }
Let SH denote the class of functions f ∈ H0, which are orientation preserving and univalent in U
An analytic function f is said to be starlike with respect to symmetric points if: z f 0 (z)

Summary

Introduction

We denote by H the class of complex-valued harmonic functions in the unit disc U := {z : |z| < r }. In 1956 Sakaguchi [1] introduced the class S ∗∗ of analytic univalent functions in U which are starlike with respect to symmetrical points. An analytic function f is said to be starlike with respect to symmetric points if:. We obtain the classes SH f ∈ SH which are convex in U (r ) or starlike in U (r ) , respectively, for any r ∈ In the present paper we obtain some analytic conditions for defined classes of functions. Some results connected to subordination properties, coefficient estimates, integral representation, and distortion theorems are obtained. (see [7,8])

Analytic Criteria

Topological Properties

Applications