Research on Geometric Mappings in Complex Systems Analysis

Yanyan Cui,Sifeng Zhu,Chaojun Wang

doi:10.1155/2016/4732704

Abstract

We mainly discuss the properties of a new subclass of starlike functions, namely, almost starlike functions of complex order λ, in one and several complex variables. We get the growth and distortion results for almost starlike functions of complex order λ. By the properties of functions with positive real parts and considering the zero of order k, we obtain the coefficient estimates for almost starlike functions of complex order λ on D. We also discuss the invariance of almost starlike mappings of complex order λ on Reinhardt domains and on the unit ball B in complex Banach spaces. The conclusions contain and generalize some known results.

Highlights

The growth, distortion theorems, and coefficient estimates for univalent functions are important research contents in geometric function theory of one and several complex variables
Let p(z) = 1 + ∑∞ n=1 cnzn be a holomorphic function on D with Rp(z) > 0
Setting λ = α/(α − 1) in Theorems 14–16, we get the corresponding results for almost starlike functions of order α

Summary

Introduction

The growth, distortion theorems, and coefficient estimates for univalent functions are important research contents in geometric function theory of one and several complex variables. Many people began to discuss the growth and distortion theorems for biholomorphic mappings with special geometric properties. In 1991, Srivastava and Owa [13] obtained the distortion theorems and coefficient estimates for a subclass S∗(α, β, γ) of starlike functions. There are many nice results about the growth, distortion theorems, and coefficient estimates for subclasses of starlike functions and convex functions. Many people generalized the Roper-Suffridge operator on different domains and different spaces so as to construct biholomorphic mappings with specific geometric properties in several complex variables. Let F(z) be a normalized locally biholomorphic mapping on the unit ball B in complex Banach spaces.

Growth and Distortion Results

Coefficient Estimates of Almost Starlike Functions of Complex Order λ

The Invariance of Almost Starlike Mappings of Complex Order λ