Dimension Free Growth Results for Vector-Valued Functions of Several Complex Variables

TL;DR

This paper establishes a dimension-free, optimal growth inequality for vector-valued entire functions of finite order less than one, extending classical results to higher dimensions with improved bounds.

Contribution

It introduces a new dimension-free inequality for the growth of vector-valued entire functions, generalizing Valiron's Theorem to several complex variables.

Findings

Derived a dimension-free growth inequality for vector-valued entire functions.

Extended Valiron's Theorem to functions in several complex variables.

Achieved best possible constants in the growth inequality.

Abstract

Let f be an entire function of finite order less than 1. The maximum modulus M(r) of f and the counting function of the zeros N(r) are connected by a best possible growth inequality known as Valiron's Theorem: For functions subharmonic in d-dimensional Euclidean space, Hayman obtained a corresponding result with a best possible constant involving the dimension d. For the special case of an entire function on d-dimensional complex space, we obtain a corresponding dimension-free, best possible inequality.