The Bohr inequality on a simply connected domain and its applications
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910433511735296 |
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| author | Ahammed, Sabir Ahamed, Molla Basir Roy, Partha Pratim |
| author_facet | Ahammed, Sabir Ahamed, Molla Basir Roy, Partha Pratim |
| contents | In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $Ω_γ,$ where $0\leq γ<1$ with a sequence $\{φ_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^{\infty}φ_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}(Ω_γ)$, where \begin{align*}
Ω_γ:=\biggl\{z\in \mathbb{C}: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ}\biggr\}. \end{align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on $ Ω_γ $. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of $ K $-quasiconformal, sense-preserving harmonic maps of the form $f=h+\overline{g}$ in $Ω_γ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01895 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Bohr inequality on a simply connected domain and its applications Ahammed, Sabir Ahamed, Molla Basir Roy, Partha Pratim Complex Variables In this article, we first establish a generalized Bohr inequality and examine its sharpness for a class of analytic functions $f$ in a simply connected domain $Ω_γ,$ where $0\leq γ<1$ with a sequence $\{φ_n(r) \}^{\infty}_{n=0}$ of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^{\infty}φ_n(r)$ converges locally uniformly on $[0,1)$. Our results represent twofold generalizations corresponding to those obtained for the classes $\mathcal{B}(\mathbb{D})$ and $\mathcal{B}(Ω_γ)$, where \begin{align*} Ω_γ:=\biggl\{z\in \mathbb{C}: \bigg|z+\dfracγ{1-γ}\bigg|<\dfrac{1}{1-γ}\biggr\}. \end{align*} As a convolution counterpart, we determine the Bohr radius for hypergeometric function on $ Ω_γ $. Lastly, we establish a generalized Bohr inequality and its sharpness for the class of $ K $-quasiconformal, sense-preserving harmonic maps of the form $f=h+\overline{g}$ in $Ω_γ.$ |
| title | The Bohr inequality on a simply connected domain and its applications |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2405.01895 |