Bohr and Rogosinski inequalities for operator valued holomorphic functions

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 ${\tilde{R}}_{p,N}(X)$ 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 $N$ ${\tilde{R}}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q(\mu )$ for $1\le p<q<\infty$ or $1\le q\le p<2$. 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 $\mathcal{H}$.

16 pages

References in corpus (1)

• The Bohr Phenomenon for analytic functions on simply connected domains

Cited by in corpus (1)

• Bohr type inequality for certain integral operators and Fourier transform on shifted disks