On a normality criterion of S. Mandelbrojt

TL;DR

This paper extends Mandelbrojt's classical normality criterion to families of holomorphic zero-free functions in several complex variables, establishing conditions based on the Levi form for normality.

Contribution

It introduces a new normality criterion for families of holomorphic functions in multiple variables using Levi form bounds, expanding classical results.

Findings

Families with Levi form bounded away from zero are normal.

Extension of Mandelbrojt's criterion to several complex variables.

Provides a geometric condition for normality.

Abstract

Extension of classical Mandelbrojt's criterion for normality of a family of holomorphic zero-free functions of several complex variables is given. We show that a family of holomorphic functions of several complex variables whose corresponding Levi form are uniformly bounded away from zero is normal.