Operator-valued analogues of multidimensional Bohr-Rogosinski inequalities

TL;DR

This paper develops operator-valued versions of multidimensional Bohr-Rogosinski inequalities, extending classical results to operator-valued functions with sharp bounds and various improvements.

Contribution

It introduces novel operator-valued multidimensional Bohr inequalities, including refined, improved, and sharp versions, expanding the scope of classical inequalities in operator theory.

Findings

Established operator-valued multidimensional refined Bohr inequality.

Proved operator-valued improved Bohr inequality with power of initial coefficient.

Derived sharp operator-valued multidimensional Bohr inequality with norm-based initial coefficient.

Abstract

In this article, we first establish operator-valued analogues of multidimensional refined Bohr inequality. Then we establish operator-valued analogues of multidimensional improved Bohr inequality with a certain power of the norm of the initial coefficient. Finally, we establish operator-valued analogues of a multidimensional sharp version of Bohr inequality with the initial coefficient being replaced by the norm value of the function. In addition, we establish operator-valued analogues of multidimensional improved Bohr inequality using the quantity $S_r/\pi$. All the results prove to be sharp.