Some Classes of Harmonic Mapping with a Symmetric Conjecture Point Defined by Subordination

Lina Ma,Xiaomeng Niu,Shuhai Li

doi:10.3390/math7060548

Abstract

In the paper, we introduce some subclasses of harmonic mapping, the analytic part of which is related to general starlike (or convex) functions with a symmetric conjecture point defined by subordination. Using the conditions satisfied by the analytic part, we obtain the integral expressions, the coefficient estimates, distortion estimates and the growth estimates of the co-analytic part g, and Jacobian estimates, the growth estimates and covering theorem of the harmonic function f. Through the above research, the geometric properties of the classes are obtained. In particular, we draw figures of extremum functions to better reflect the geometric properties of the classes. For the first time, we introduce and obtain the properties of harmonic univalent functions with respect to symmetric conjugate points. The conclusion of this paper extends the original research.

Highlights

Introduction and PreliminariesLet A denote the class of functions in the following form ∞ h(z) = z + ∑ an zn, (1)n =2 where h(z) is analytic in the open unit disk U = {z ∈ C : |z| < 1}.S, S ∗, K are denoted respectively by the subclasses of A consisting of univalent, starlike, convex functions.Let P denote the class of functions p satisfying p(0) = 1 and Rep(z) > 0, where z ∈ U.The function s is subordinate to t in U, written by s(z) ≺ t(z), if there exists a Schwarz function σ, analytic in U with σ (0) = 0 and |σ (z)| < 1, satisfying s(z) = t(σ (z))(see [1])
In 1994, Ma and Minda [3] introduce a class S ∗ (φ) of starlike functions defined by subordination, zh0 (z) h(z) ∈ S ∗ (φ) if and only if h(z) ≺ φ(z), where h ∈ A, φ ∈ P
In 1987, El-Ashwa and Thomas [7] introduced some classes of starlike functions with respect to conjugate points and symmetric conjugate points satisfying the following conditions

Summary

Let A denote the class of functions in the following form

In 1959, Sakaguchi [6] introduced the class Ss∗ of starlike functions with respect to symmetric points, f ∈ Ss∗ if and only if z f 0 (z). In 1987, El-Ashwa and Thomas [7] introduced some classes of starlike functions with respect to conjugate points and symmetric conjugate points satisfying the following conditions z f 0 (z). Zhu and Huang [17] studied some subclasses of S H with h is convex, or starlike functions of order β and some sharp estimates of coefficients, distortion, and growth are obtained. According to the principle of subordination, we introduce the following general subclasses of S H of harmonic univalent starlike and convex functions with a symmetric conjecture point.

Scs cs cs cs sc sc sc sc cs cs

Main Results

It is easy to show that