Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913237048492032 |
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| author | Ahamed, Molla Basir Roy, Partha Pratim |
| author_facet | Ahamed, Molla Basir Roy, Partha Pratim |
| contents | Let $ \mathcal{H}(Ω) $ be the class of complex-valued functions harmonic in $ Ω\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(Ω)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $ r_f\in (0, 1) $ such that the inequality
\begin{align*}
M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f,
\end{align*}
where $ d\left(f(0), \partialΩ\right) $ is the Euclidean distance between $ f(0) $ and the boundary of $ Ω:=f(\mathbb{D}) $. The largest such radius $ r_f $ is called the Bohr radius and the inequality $ M_f(r)\leq d\left(f(0), \partialΩ\right) $ is called the Bohr inequality for the class $ \mathcal{H}(Ω) $. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11808 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings Ahamed, Molla Basir Roy, Partha Pratim Complex Variables Primary 30C45, 30C50, 30C80 Let $ \mathcal{H}(Ω) $ be the class of complex-valued functions harmonic in $ Ω\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(Ω)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $ r_f\in (0, 1) $ such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where $ d\left(f(0), \partialΩ\right) $ is the Euclidean distance between $ f(0) $ and the boundary of $ Ω:=f(\mathbb{D}) $. The largest such radius $ r_f $ is called the Bohr radius and the inequality $ M_f(r)\leq d\left(f(0), \partialΩ\right) $ is called the Bohr inequality for the class $ \mathcal{H}(Ω) $. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp. |
| title | Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings |
| topic | Complex Variables Primary 30C45, 30C50, 30C80 |
| url | https://arxiv.org/abs/2402.11808 |