Multidimensional analogues of the refined Bohr type inequalities

TL;DR

This paper extends classical Bohr inequalities to multiple complex variables, providing sharp bounds for bounded holomorphic functions in polydisks and their derivatives, advancing the understanding of multidimensional complex analysis.

Contribution

It introduces sharp multidimensional analogues of Bohr inequalities, including variants with absolute values and derivatives, for functions in polydisks, filling gaps in the theory of several complex variables.

Findings

Established sharp multidimensional Bohr inequalities in polydisks.

Proved variants replacing the constant term with the absolute value and its square.

Extended known results to the setting of several complex variables.

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 $\mathbb{D}^n$.We also prove two other sharp versions of the Bohr inequality in the setting of several complex variables: one by replacing the constant term with the absolute value of the function, and another by replacing it with the square of the absolute value of the function.Furthermore, we establish multidimensional analogues of known results concerning the modulus of the derivative of analytic functions in the unit disk $\mathbb{D}$, replacing the derivative with the radial derivative of holomorphic functions in $\mathbb{D}^n$.All of the established results are shown to be sharp.