A Landau's theorem in several complex variables

TL;DR

This paper extends Landau's theorem to certain classes of holomorphic maps in several complex variables, establishing universal radius results and Brody-Zalcman theorems where counterexamples previously existed.

Contribution

It introduces a new class of holomorphic maps in several complex variables for which Landau's theorem and Brody-Zalcman theorems hold.

Findings

Established a Landau-type theorem for specific holomorphic maps in multiple variables.

Proved a Brody-Zalcman theorem in several complex variables for the introduced class.

Identified conditions under which universal radius results are valid in higher dimensions.

Abstract

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $\rho$, independent of the functions, such that in the image of the unit disc of any of the functions of the family there is a disc of universal radius $\rho.$ This is the so celebrated Landau's theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper we introduce a class of holomorphic maps for which one can get a Landau's theorem and a Brody-Zalcman theorem in several complex variables.