Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n

TL;DR

This paper extends Bohr-type inequalities to several complex variables, determining sharp radii for holomorphic functions on the polydisc and analyzing directional derivatives to generalize univariate results.

Contribution

It provides the first definitive solution to the Bohr phenomenon in multiple complex variables, including sharp radii and growth estimates for multidimensional Schwarz functions.

Findings

Determined sharp Bohr radii for bounded holomorphic functions in ^n

Established refined growth estimates for directional derivatives in ^n

Proved the optimality of all obtained constants

Abstract

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}\Delta(0;1_n)$. We provide a definitive resolution to the Bohr phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $\omega_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the directional derivative operator $\partial_uf(z) = \sum_{k=1}^{n} u_k \frac{\partial f(z)}{\partial z_k}$, where $u=(u_1,u_2,\ldots,u_n)\in\mathbb{C}^n$ such that $|u_1|+|u_2|+\ldots+|u_n|=1$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.