Multidimensional analogues of the refined versions of Bohr inequalities involving Schwarz mappings

TL;DR

This paper develops new refined Bohr inequalities for bounded analytic functions and extends these results to multidimensional settings involving Schwarz mappings, with all findings proven to be sharp.

Contribution

It introduces novel refined Bohr inequalities involving Schwarz functions and extends these to multidimensional complex Banach spaces with sharp results.

Findings

New refined Bohr inequalities for bounded analytic functions

Multidimensional analogues involving higher-dimensional Schwarz mappings

Results are proven to be sharp

Abstract

Our first aim of this article is to establish several new versions of refined Bohr inequalities for bounded analytic functions in the unit disk involving Schwarz functions. Secondly, %as applications of these results, we obtain several new multidimensional analogues of the refined Bohr inequalities for bounded holomorphic mappings on the unit ball in a complex Banach space involving higher dimensional Schwarz mappings. All the results are proved to be sharp.