Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n
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arXiv
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| Natura: | Preprint |
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2026
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| author | Ahamed, Molla Basir Majumder, Sujoy Pramanik, Debabrata |
| author_facet | Ahamed, Molla Basir Majumder, Sujoy Pramanik, Debabrata |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n Ahamed, Molla Basir Majumder, Sujoy Pramanik, Debabrata Complex Variables Primary 32A05, 30C80, Secondary 32A10, 41A17 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. |
| title | Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n |
| topic | Complex Variables Primary 32A05, 30C80, Secondary 32A10, 41A17 |
| url | https://arxiv.org/abs/2603.03349 |