Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n
Molla Basir Ahamed, Sujoy Majumder, Nabadwip Sarkar
TL;DR
The paper advances Bohr-type inequalities to the polydisk $\mathbb{D}^n$, proving that the sharp Bohr radius remains $R_n=\frac{1}{3n}$ for holomorphic maps $f:\mathbb{D}^n\to\mathbb{D}$ and delivering a multidimensional resolution of the Bohr–Rogosinski phenomenon via radii $R_{m,n,N}$ tied to $\psi_{m,n,N}$. It also extends derivative growth estimates using the radial Euler operator $Df(z)$ and establishes a multidimensional area-based Bohr inequality, all with sharp constants. The approach hinges on multivariable coefficient bounds, Schwarz-type estimates, and carefully constructed extremals to verify optimality. Together, these results unify and extend univariate Bohr theory to several complex variables, providing sharp, geometry-aware bounds that reflect the polydisc structure and its Schwarz-class mappings.
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 $ω_{n,m}\in\mathcal{B}_{n,m}$ 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.