Zalcman Conjecture for Starlike Mappings in Higher Dimensions

TL;DR

This paper extends the Zalcman conjecture to higher dimensions, providing sharp bounds for starlike mappings in complex spaces and confirming the conjecture's validity for three dimensions.

Contribution

It establishes sharp bounds for the Zalcman functional in three dimensions for starlike mappings, advancing geometric function theory in several complex variables.

Findings

Confirmed the Zalcman conjecture for n=3 in higher dimensions.

Derived sharp bounds for the Zalcman functional in complex Banach spaces.

Extended results to the unit polydisk in ^n.

Abstract

Counterexamples show that many results in the geometric function theory of one complex variable are not applicable for several complex variables. In this paper, we obtain sharp bounds for the Zalcman functional for $n=3$ associated with the starlike mappings defined on the unit ball in a complex Banach space and on the unit polydisk in $\mathbb{C}^n$. These results confirm the validity of the Zalcman conjecture in higher dimensions for $n=3$.