Bohr and Rogosinski inequalities for operator valued holomorphic functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913027429761024 |
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| author | Allu, Vasudevarao Halder, Himadri Pal, Subhadip |
| author_facet | Allu, Vasudevarao Halder, Himadri Pal, Subhadip |
| contents | For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each $N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly $\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order $N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q (μ)$ for $1\leq p<q<\infty$ or $1\leq q \leq p <2$. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk $\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$ denotes the space of all bounded linear operator on a complex Hilbert space $\mathcal{H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_03849 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bohr and Rogosinski inequalities for operator valued holomorphic functions Allu, Vasudevarao Halder, Himadri Pal, Subhadip Functional Analysis 46E40, 47A56, 46B20, 47A63, 30B10 For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each $N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly $\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order $N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q (μ)$ for $1\leq p<q<\infty$ or $1\leq q \leq p <2$. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk $\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$ denotes the space of all bounded linear operator on a complex Hilbert space $\mathcal{H}$. |
| title | Bohr and Rogosinski inequalities for operator valued holomorphic functions |
| topic | Functional Analysis 46E40, 47A56, 46B20, 47A63, 30B10 |
| url | https://arxiv.org/abs/2201.03849 |