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

Abstract

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}\hat{I}”(0;1_n)$. 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 $\ddot{I}{‰}_{n,m}\in {\mathcal{B}}_{n,m}$ and the local modulus $|f(z)|$. By employing the directional derivative operator ${\partial }_{u}f(z)={\sum }_{k=1}^n{u}_k\frac{\partial f(z)}{\partial z_k}$, where $u=(u_1,u_2,\dots ,u_n)\in {\mathbb{C}}^n$ such that $|u_1|+|u_2|+\dots +|u_n|=1$, 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.

21 pages. arXiv admin note: substantial text overlap with arXiv:2601.06630