{ "id": "2201.03849", "version": "v2", "published": "2022-01-11T09:17:22.000Z", "updated": "2022-01-19T11:38:34.000Z", "title": "A Geometric characterization of Banach spaces with $p$-Bohr radius", "authors": [ "Vasudevarao Allu", "Himadri Halder", "Subhadip Pal" ], "comment": "9 pages", "categories": [ "math.FA" ], "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 $f(z)=\\sum_{k=0}^{\\infty} x_{k}z^k \\in H^{\\infty}(\\mathbb{D}, X)$. We also introduce the following geometric notion of $p$-uniformly $\\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \\in \\mathbb{N}$. For $p\\in [2,\\infty)$, a complex Banach space $X$ is called $p$-uniformly $\\mathbb{C}$-convex of order $N$ if there exists a constant $\\lambda > 0$ such that \\begin{align}\\label{e-0.1} \\left(\\norm{x_0}^p + \\lambda \\norm{x_1}^p + {\\lambda}^2 \\norm{x_2}^p + \\cdots + {\\lambda}^N \\norm{x_N}^p \\right)^{1/p} \\leq \\max_{\\theta \\in [0,2\\pi)} \\norm{x_0 + \\sum_{k=1}^{N}e^{i \\theta}x_k} \\end{align} for all $x_0$, $x_1$,$\\dots$, $x_N$ $\\in X$. We denote $A_{p,N}(X)$, the supremum of all such constants $\\lambda$ satisfying \\eqref{e-0.1}. We obtain the lower and upper bounds of $\\widetilde{R}_{p,N}(X)$ in terms of $A_{p,N}(X)$. 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 $L^q (\\mu)$ and the sequence spaces $l^q$ for $1\\leq p