Multidimensional analogues of the improved Bohr's inequality

TL;DR

This paper extends the classical Bohr inequality to multiple complex variables, providing sharp bounds for bounded holomorphic functions in polydisks with novel versions involving absolute values.

Contribution

It introduces multidimensional sharp improvements of Bohr's inequality, including variants with absolute values and squared absolute values, in several complex variables.

Findings

Established sharp bounds for holomorphic functions in polydisks.

Proved two new sharp versions of the Bohr inequality involving absolute values.

Results are optimal and extend classical inequalities to higher dimensions.

Abstract

The main aim of this article is to establish a sharp improvement of the classical Bohr inequality for bounded holomorphic mappings in the polydisk $\mathbb{P}\Delta(0;1_n)$. We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables by replacing the constant term with the absolute value of the function and the square of the absolute value of the function, respectively. All the results are shown to be sharp.