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Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables

Surya Giri

Abstract

Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$.

Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables

Abstract

Generalizing the Zalcman conjecture given by , Ma proposed and proved that the inequality holds for functions , the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for and , considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in .
Paper Structure (2 sections, 6 theorems, 53 equations)

This paper contains 2 sections, 6 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Lemma 1

Suf Let $f: \mathbb{B} \rightarrow X$ be a normalized locally biholomorphic mapping. The mapping $f$ is said to be starlike on $\mathbb{B}$ if and only if The class of all such mappings on $\mathbb{B}$ is denoted by $\mathcal{S}^*(\mathbb{B})$.

Theorems & Definitions (10)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof
  • Remark
  • Theorem 6
  • proof
  • Remark