Bohr phenomena for slice regular functions over Quaternions

TL;DR

This paper extends the Bohr inequality to slice regular functions over quaternions, providing new bounds and refinements for classes like starlike and close-to-convex functions, with sharp results.

Contribution

It establishes the Bohr inequality for quaternionic slice regular functions, generalizes and improves existing bounds, and offers sharp versions for specific function classes.

Findings

Proved Bohr inequality for slice starlike functions over quaternions.

Derived improved versions of the Bohr inequality for slice regular functions.

Provided sharp bounds for functions with real part bounded by 1.

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.