Bohr phenomena for slice regular functions over Quaternions

Abstract

Slice regular functions are a generalization of holomorphic functions to the setting of quaternions (and more generally, Clifford algebras). In this paper, we first establish the Bohr inequality for slice starlike functions and slice close-to-convex functions over quaternions $\mathbb{H}$. Next, we present a generalization of the Bohr inequality, and improved versions of the Bohr inequality for slice regular functions on the open unit ball $\mathbb{B}$ of $\mathbb{H}$. Finally, we provide a refined version of the Bohr inequality for slice regular functions $f$ on $\mathbb{B}$ such that $ {\rm Re}(f(q)) \leq 1 $ for all $q \in \mathbb{B}$. All the results are demonstrated to be sharp.