Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n
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| Format: | Preprint |
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2026
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| _version_ | 1866915720441364480 |
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| author | Ahamed, Molla Basir Majumder, Sujoy Sarkar, Nabadwip |
| author_facet | Ahamed, Molla Basir Majumder, Sujoy Sarkar, Nabadwip |
| contents | This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{D}^n$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_06630 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n Ahamed, Molla Basir Majumder, Sujoy Sarkar, Nabadwip Complex Variables 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 $\mathbb{D}^n$. 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. |
| title | Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n |
| topic | Complex Variables 32A05, 30C80, Secondary 32A10, 41A17 |
| url | https://arxiv.org/abs/2601.06630 |