Bohr radius and its asymptotic value for holomorphic functions in higher dimensions

Abstract

We establish sharp Bohr phenomena for holomorphic functions defined on a bounded balanced domain $G$ in a complex Banach space $X$, which map into a simply connected domain or a convex domain $\Omega $ in the complex plane $\mathbb{C}$. Taking $X$ as the $n$-dimensional complex plane and $G$ as the open unit polydisk, we consider a version of the Bohr inequality stronger than the above mentioned one and study the exact asymptotic behaviour of the Bohr radius. Explicit lower bounds on the Bohr radii of this type are also provided. Extending a recent result of Liu and Ponnusamy, we further record a refined form of the Bohr inequality for the particular case $\Omega =\mathbb{D}$, i.e. the open unit disk in $\mathbb{C}$.