Bohr radius for Banach spaces on simply connected domains (2111.10880v2)

Abstract: Let $H^{\infty}(\Omega,X)$ be the space of bounded analytic functions $f(z)=\sum_{n=0}^{\infty} x_{n}z^{n}$ from a proper simply connected domain $\Omega$ containing the unit disk $\mathbb{D}:={z\in \mathbb{C}:|z|<1}$ into a complex Banach space $X$ with $\norm{f}{H^{\infty}(\Omega,X)} \leq 1$. 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 $1\leq p,q<\infty$, we denote \begin{equation*} R_{p,q,\phi}(f,\Omega,X)= \sup \left{r \geq 0: \norm{x_{0}}^p \phi_{0}(r) + \left(\sum_{n=1}^{\infty} \norm{x_{n}}\phi_{n}(r)\right)^q \leq \phi_{0}(r)\right} \end{equation*} and define the Bohr radius associated with $\phi$ by $$R_{p,q,\phi}(\Omega,X)=\inf \left{R_{p,q,\phi}(f,\Omega,X): \norm{f}{H^{\infty}(\Omega,X)} \leq 1\right}.$$ In this article, we extensively study the Bohr radius $R{p,q,\phi}(\Omega,X)$, when $X$ is an arbitrary Banach space and $X$ is certain Hilbert space. Furthermore, we establish the Bohr inequality for the operator-valued Ces\'{a}ro operator and Bernardi operator.