Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables

Abstract

Generalizing the Zalcman conjecture given by $|a_n^2-a_{2n-1}|\le (n-1{)}^2$, Ma proposed and proved that the inequality $$|a_n{a}_m-a_{n+m-1}|\le (n-1)(m-1),m,n\in \mathbb{N},$$ holds for functions $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots \in {\mathcal{S}}^{\ast }$, 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 ${\mathbb{C}}^n$.