Refined Bohr inequalities and a refined Bohr-Rogosinski inequality on complex Banach spaces

TL;DR

This paper develops refined and sharp versions of classical Bohr and Bohr-Rogosinski inequalities for holomorphic functions on the unit ball of complex Banach spaces, with applications to lacunary series and higher-dimensional mappings.

Contribution

It introduces new refined inequalities for holomorphic functions on Banach space unit balls, extending classical results with sharp bounds and applications.

Findings

Refined Bohr inequalities for holomorphic functions on Banach spaces

Bohr-Rogosinski inequality established for these functions

Improved and sharp bounds for holomorphic functions on $B_X$

Abstract

In this paper, we first establish refined versions of the Bohr inequalities for the class of holomorphic functions from the unit ball $B_X$ of a complex Banach space $X$ into $\mathbb{C}$. As applications, we will establish refined Bohr inequalities of functional type or of norm type for holomorphic mappings with lacunary series on the unit ball $B_X$ with values in higher dimensional spaces. Next, we obtain the Bohr-Rogosinski inequality for the class of holomorphic functions on $B_X.$ In addition, we establish an improved version of the Bohr inequality for holomorphic functions on $B_X$. All the results are proved to be sharp.