On the Fekete and Szegö Problem for the Class of Starlike Mappings in Several Complex Variables

Qing-Hua Xu,Tai-Shun Liu

doi:10.1155/2014/807026

Abstract

LetSbe the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known resultmaxf∈S⁡a3-λa22=1+2e-2λ/(1-λ)forλ∈0, 1. We investigate the corresponding problem for the class of starlike mappings defined on the unit ball in a complex Banach space or on the unit polydisk inCn, which satisfies a certain condition.

Highlights

Let A be the class of functions of the form ∞f (z) = z + ∑anzn, (1)n=2 which are analytic in the open unit diskU = {z ∈ C : |z| < 1} . (2)We denote by S the subclass of the normalized analytic function class A consisting of all functions which are univalent in U
A holomorphic mapping f : B → X is said to be biholomorphic if the inverse f−1 exists and is holomorphic on the open set f(B)
The following example shows that the estimation of Theorem 1 is sharp

Summary

Introduction

Let H(B) denote the set of all holomorphic mappings from B into X. A holomorphic mapping f : B → X is said to be biholomorphic if the inverse f−1 exists and is holomorphic on the open set f(B). A mapping f ∈ H(B) is said to be locally biholomorphic if the Frechet derivative Df(x) has a bounded inverse for each x ∈ B. Let S(B) be the set of all normalized biholomorphic mappings on B. Let S∗(B) be the set of normalized starlike mappings on B.

Results

Conclusion