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arXiv · 2603.03349

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

Molla Basir Ahamed·Sujoy Majumder·Debabrata Pramanik
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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.

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Molla Basir Ahamed, Sujoy Majumder, Debabrata Pramanik. 2026-02-26. Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n. https://arxiv.org/abs/2603.03349

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