Bohr and Rogosinski inequalities for operator valued holomorphic functions

TL;DR

This paper introduces the $p$-Bohr radius and $p$-uniformly $ ext{C}$-convexity for Banach spaces, establishing their equivalence for certain $p$, and extends classical inequalities to operator-valued holomorphic functions.

Contribution

It defines new geometric and radius concepts for Banach spaces and proves their equivalence, also extending Bohr and Rogosinski inequalities to operator-valued functions.

Findings

The $p$-Bohr radius is positive if and only if the space is $p$-uniformly $ ext{C}$-convex of order $N$.

The paper characterizes the $p$-Bohr radius for Lebesgue spaces $L^q( u)$.

An operator-valued analogue of Bohr and Rogosinski inequalities is established.

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 $\mathcal{H}$.