Bohr radius for Banach spaces on simply connected domains
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866911589408440320 |
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| author | Allu, Vasudevarao Halder, Himadri |
| author_facet | Allu, Vasudevarao Halder, Himadri |
| contents | Let $H^{\infty}(Ω,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $Ω$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ into a complex Banach space $X$ with $\norm{f}_{H^{\infty}(Ω,X)} \leq 1$. Let $ϕ=\{ϕ_{n}(r)\}_{n=0}^{\infty}$ with $ϕ_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} ϕ_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q<\infty$, we denote
\begin{equation*}
R_{p,q,ϕ}(f,Ω,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p ϕ_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}ϕ_{n}(r)\right)^q \leq ϕ_{0}(r)\right\}
\end{equation*}
and define the Bohr radius associated with $ϕ$ by $$R_{p,q,ϕ}(Ω,X)=\inf \left\{R_{p,q,ϕ}(f,Ω,X): \norm{f}_{H^{\infty}(Ω,X)} \leq 1\right\}.$$
In this article, we extensively study the Bohr radius $R_{p,q,ϕ}(Ω,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10880 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bohr radius for Banach spaces on simply connected domains Allu, Vasudevarao Halder, Himadri Complex Variables Functional Analysis 46E40, 47A56, 47A63, 46B20, 30B10, 30C20, 30C65 Let $H^{\infty}(Ω,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $Ω$ containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ into a complex Banach space $X$ with $\norm{f}_{H^{\infty}(Ω,X)} \leq 1$. Let $ϕ=\{ϕ_{n}(r)\}_{n=0}^{\infty}$ with $ϕ_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} ϕ_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q<\infty$, we denote \begin{equation*} R_{p,q,ϕ}(f,Ω,X)= \sup \left\{r \geq 0: \norm{x_{0}}^p ϕ_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}ϕ_{n}(r)\right)^q \leq ϕ_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with $ϕ$ by $$R_{p,q,ϕ}(Ω,X)=\inf \left\{R_{p,q,ϕ}(f,Ω,X): \norm{f}_{H^{\infty}(Ω,X)} \leq 1\right\}.$$ In this article, we extensively study the Bohr radius $R_{p,q,ϕ}(Ω,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator. |
| title | Bohr radius for Banach spaces on simply connected domains |
| topic | Complex Variables Functional Analysis 46E40, 47A56, 47A63, 46B20, 30B10, 30C20, 30C65 |
| url | https://arxiv.org/abs/2111.10880 |