Bohr radius for Banach spaces on simply connected domains

Abstract

AbstractLet $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain Ω containing the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z| \lt 1\}$ into a complex Banach space X with $\left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1$. Let $\phi=\{\phi_{n}(r)\}_{n=0}^{\infty}$ with $\phi_{0}(r)\leq 1$ such that $\sum_{n=0}^{\infty} \phi_{n}(r)$ converges locally uniformly with respect to $r \in [0,1)$. For $1\leq p,q \lt \infty$, we denote \begin{equation*} R_{p,q,\phi}(f,\Omega,X)= \sup \left\{r \geq 0: \left\lVert x_{0}\right\rVert^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \left\lVert x_{n}\right\rVert\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right\} \end{equation*} and define the Bohr radius associated with ϕ by \begin{equation*}R_{p,q,\phi}(\Omega,X)=\inf \left\{R_{p,q,\phi}(f,\Omega,X): \left\lVert f\right\rVert_{H^{\infty}(\Omega,X)} \leq 1\right\}.\end{equation*} In this article, we extensively study the Bohr radius $R_{p,q,\phi}(\Omega,X)$, when X is an arbitrary Banach space, and $X=\mathcal{B}(\mathcal{H})$ is the algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$. Furthermore, we establish the Bohr inequality for the operator-valued Cesáro operator and Bernardi operator.