Multidimensional analogues of refined Bohr's inequality

TL;DR

This paper develops multidimensional versions of refined and improved Bohr's inequalities, extending classical results to higher dimensions and demonstrating the sharpness of most results.

Contribution

It introduces new multidimensional analogues of refined and improved Bohr's inequalities, including cases with initial coefficients and absolute value replacements.

Findings

Established multidimensional analogues of refined Bohr's inequality.

Proved two versions of improved Bohr's inequality with zero initial coefficient.

Demonstrated the sharpness of most results.

Abstract

In this paper, we first establish a version of multidimensional analogues of the refined Bohr's inequality. Then we establish two versions of multidimensional analogues of improved Bohr's inequality with initial coefficient being zero. Finally we establish two versions of multidimensional analogues of improved Bohr's inequality with the initial coefficient being replaced by absolute value of the function, and to prove that most of the results are sharp.