Bohr inequalities for holomorphic mappings in higher-dimensional complex Banach spaces

TL;DR

This paper extends Bohr inequalities to holomorphic mappings in higher-dimensional complex Banach spaces, providing sharper bounds and analyzing compositions with Schwarz mappings, with all results proven to be optimal.

Contribution

It introduces improved and refined Bohr inequalities for holomorphic maps in Banach spaces, including composition cases, establishing sharp Bohr radii.

Findings

Derived improved Bohr inequality with squared norms

Established refined Bohr inequality involving coefficient norms

Proved sharpness of the Bohr radius in each case

Abstract

In this paper, we investigates the Bohr phenomenon for holomorphic mappings $F$ from the unit ball $\mathbb{B}_X$ of a complex Banach space $X$ into the closure of the unit polydisc $\mathbb{D}^m$ within the space $\mathbb{C}^m$. First, we prove an improved Bohr inequality involving the squared norms of the mapping and its homogeneous expansions. Second, we derive a refined Bohr inequality that incorporates a combination of the coefficient norms and their squares. Finally, we obtain a refined Bohr inequality for compositions $F\circ \nu$, where $\nu$ is a Schwarz mapping with a zero of order $k$ at the origin. For each result, we demonstrate that the derived Bohr radius is sharp.