Bohr and Rogosinski inequalities for operator valued holomorphic functions

[2201.03849] Bohr and Rogosinski inequalities for operator valued holomorphic functions Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Functional Analysis arXiv:2201.03849 (math) [Submitted on 11 Jan 2022 ( v1 ), last revised 21 Sep 2022 (this version, v3)] Title: Bohr and Rogosinski inequalities for operator valued holomorphic functions Authors: Vasudevarao Allu , Himadri Halder , Subhadip Pal View a PDF of the paper titled Bohr and Rogosinski inequalities for operator valued holomorphic functions, by Vasudevarao Allu and 2 other authors View PDF HTML (experimental) Abstract: 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 $L^q (\mu)$ 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 $\mathcal{H}$. Comments: 16 pages Subjects: Functional Analysis (math.FA) MSC classes: 46E40, 47A56, 46B20, 47A63, 30B10 Cite as: arXiv:2201.03849 [math.FA] (or arXiv:2201.03849v3 [math.FA] for this version) https://doi.org/10.48550/arXiv.2201.03849 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Bulletin des Sciences Mathématiques 182 (2023) 103214 Related DOI : https://doi.org/10.1016/j.bulsci.2022.103214 Focus to learn more DOI(s) linking to related resources Submission history From: Himadri Halder [ view email ] [v1] Tue, 11 Jan 2022 09:17:22 UTC (17 KB) [v2] Wed, 19 Jan 2022 11:38:34 UTC (17 KB) [v3] Wed, 21 Sep 2022 18:21:14 UTC (21 KB) Full-text links: Access Paper: View a PDF of the paper titled Bohr and Rogosinski inequalities for operator valued holomorphic functions, by Vasudevarao Allu and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.FA < prev | next > new | recent | 2022-01 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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