Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables

TL;DR

This paper extends a classical inequality related to starlike functions from one complex variable to several complex variables, involving mappings on complex Banach spaces and circular domains.

Contribution

It generalizes the Zalcman conjecture for starlike functions to multiple complex variables and different domain types.

Findings

Established inequality for multivariable starlike mappings

Extended classical conjecture to complex Banach spaces

Analyzed mappings on circular domains in $ extbf{C}^n$

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 $\mathbb{C}^n$.