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

Molla Basir Ahamed, Sujoy Majumder, Debabrata Pramanik

Abstract

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}Δ(0;1_n)$. 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 $ω_{n,m}\in\mathcal{B}_{n,m}$ 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.

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 . 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 and the local modulus . By employing the directional derivative operator , where such that , we obtain refined growth estimates for derivatives that generalize well-known univariate results to . The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.
Paper Structure (6 sections, 9 theorems, 105 equations)

This paper contains 6 sections, 9 theorems, 105 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Classical Bohr inequality and its recent implications
  3. Basic Notations in several complex variables
  4. Main Results
  5. Key lemmas
  6. Proofs of the main results

Key Result

Theorem 2.1

Let $f(z)=\sum_{|\alpha|=0} a_{\alpha}z^{\alpha}$ be a holomorphic function in the polydisk $\mathbb{P}\Delta(0;1_n)$ such that $|f(z)|\leq 1$ for all $z\in \mathbb{P}\Delta(0;1/n)$. Suppose $z=(z_1,\ldots,z_n)\in \mathbb{P}\Delta(0;1/n)$ and $r=(r_1,r_2,\ldots,r_n)$ such that $||z||_{\infty}={\bf r for ${\bf r}\leq R_{m,n,t}$, where The radius $R_{m,n,t}$ is best possible.

Theorems & Definitions (16)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 6 more