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

TL;DR

The paper extends the Bohr phenomenon to holomorphic mappings from the unit ball $\mathbb{B}_X$ of a complex Banach space into the closed polydisc $\overline{\mathbb{D}^m}$, using homogeneous polynomial expansions to derive sharp Bohr-type inequalities. By combining coefficient-norm bounds with Fréchet-derivative expansions, the authors establish explicit Bohr radii, including $\sqrt{11/27}$ for a squared-norm Bohr inequality and refined radii $\tfrac{1}{5}$ and $\tfrac{1}{3}$ for additional coefficient terms. They further generalize to compositions with Schwarz mappings of order $k$, obtaining a radius $r_k$ given by the root of $\frac{1-r^{k}}{1+r^k}-\frac{2r}{1-r}=0$, with sharpness demonstrated via extremal examples. The results deepen understanding of Bohr-type phenomena in asymmetric, Banach-space-valued holomorphic settings and relate to Bohr-Rogosinski-type considerations in higher dimensions.

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 $F\circ ν$, where $ν$ 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.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• proof
• Example 3.1
• Theorem 3.2
• proof