Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n

TL;DR

This paper extends classical Bohr inequalities to multiple complex variables, establishing sharp radii for bounded holomorphic functions, Schwarz functions, and growth estimates using Euler derivatives in , with all constants proven optimal.

Contribution

It provides the first sharp multidimensional Bohr and Bohr-Rogosinski radii, generalizes growth estimates with Euler derivatives, and introduces a multidimensional area-based Bohr inequality.

Findings

Bohr radius remains 1/(3n) in  for bounded holomorphic functions.

Sharp radii are determined for Schwarz functions and local modulus in several complex variables.

Multidimensional growth estimates and area-based inequalities are established with optimal 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 $\mathbb{D}^n$. We establish a sharp extension of the classical Bohr inequality, proving that the Bohr radius remains $R_n = 1/(3n)$ for the family of holomorphic functions bounded by unity in the multivariate setting. Further, we provide a definitive resolution to the Bohr-Rogosinski 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 radial (Euler) derivative operator $Df(z) = \sum_{k=1}^{n} z_k \frac{\partial f(z)}{\partial z_k}$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. Finally, a multidimensional version of the area-based Bohr inequality is established. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.