Zalcman Conjecture for Starlike Mappings in Higher Dimensions

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 $n=3$.