The Bohr radius for operator valued functions on simply connected domain

Abstract

In this paper, we first establish an improved Bohr inequality for the class of operator-valued holomorphic functions $f$ on a simply connected domain $\hat{I}\text{c◯}$ in $\mathbb{C}$. Next, we establish a generalization of refined version of the Bohr inequality and the Bohr-Rogosinski inequality with the help of the sequence $\ddot{I}†=\{\ddot{I}{†}_n(r){\}}_{n=0}^{\infty }$ of non-negative continuous functions in $[0,1)$ such that the series ${\sum }_{n=0}^{\infty }\ddot{I}{†}_n(r)$ converges locally uniformly on the interval $[0,1)$. All the results are proved to be sharp. Moreover, We establish the Bohr inequality and the Bohr-Rogosinski inequality for the class of operator-valued $\hat{I}½$-Bloch functions defined in two different simply connected domains, $\hat{I}\text{c◯}$ and $\hat{I}{\text{c◯}}_{\hat{I}}^3$, in $\mathbb{C}$.