Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk
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| Format: | Preprint |
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2024
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| _version_ | 1866910343145455616 |
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| author | Ahamed, Molla Basir Roy, Partha Pratim |
| author_facet | Ahamed, Molla Basir Roy, Partha Pratim |
| contents | In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks $\mathbb{D}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0, 1, 2, 3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{φ_n(r)\}_{n=0}^{\infty}$ of continuous functions of $r$ in $[0, 1)$ and give some applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk Ahamed, Molla Basir Roy, Partha Pratim Complex Variables Primary 30A10, 30H05, 30C35, 30C50 Secondary 30C45 In this paper, we study some improved and refined versions of the classical Bohr inequality applicable to the class $\mathcal{B}$, which consists of self-analytic mappings defined on the unit disk $\mathbb{D}$. First, we improve the Bohr inequality for the class $\mathcal{B}$ of analytic self-maps, incorporating the area measurements of sub-disks $\mathbb{D}_r$ of $\mathbb{D}$. Secondly, we establish a sharp inequality with suitable setting as an improved version of the classic Bohr inequality. Then we obtain a sharp refined Bohr inequality in which the coefficients $|a_k|$ $(k=0, 1, 2, 3)$ in the majorant series $M_f(r)$ of $f$ are replaced by $|f^{(k)}(z)|/k!$. Finally, for a certain class $\mathcal{P}^0_{\mathcal{H}}(M)$ of harmonic mappings of the form $f=h+\overline{g}$, we generalize the Bohr inequality incorporating a sequence $\{φ_n(r)\}_{n=0}^{\infty}$ of continuous functions of $r$ in $[0, 1)$ and give some applications. |
| title | Revisiting Bohr Inequalities with Analytic and Harmonic Mappings on unit disk |
| topic | Complex Variables Primary 30A10, 30H05, 30C35, 30C50 Secondary 30C45 |
| url | https://arxiv.org/abs/2402.15689 |