A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908614021611520 |
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| author | Ahamed, Molla Basir Roy, Partha Pratim Majumder, Sujoy |
| author_facet | Ahamed, Molla Basir Roy, Partha Pratim Majumder, Sujoy |
| contents | We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(α)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{φ_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_22509 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk Ahamed, Molla Basir Roy, Partha Pratim Majumder, Sujoy Complex Variables Primary 30A10, 30H05, 30C35, Secondary 30C45 We investigate improved forms of the Bohr inequality, using the quantity $S_r/π$, for analytic selfmaps in class $\mathcal{B}$ of $\mathbb{D}$, where $S_r$ is the area measure of $\mathbb{D}_r$. We then generalize the inequality for harmonic mappings ($\mathcal{P}^0_{\mathcal{H}}(M)$ and $\mathcal{W}^0_{\mathcal{H}}(α)$ of the form $f = h + \overline{g}$) by introducing a sequence $\{φ_n(r)\}_{n=0}^\infty$ of differentiable, increasing functions on $[0, 1)$. The Hurwitz Lerch Zeta function is utilized for some consequences, and all results are shown to be sharp. |
| title | A Unified Study of Bohr's Inequality for analytic and harmonic mappings on the Unit Disk |
| topic | Complex Variables Primary 30A10, 30H05, 30C35, Secondary 30C45 |
| url | https://arxiv.org/abs/2510.22509 |