On two Bloch type theorems for quaternionic slice regular functions

TL;DR

This paper establishes two Bloch type theorems for quaternionic slice regular functions, analyzing their injective and covering properties, and demonstrating the existence of a universal ball in their images.

Contribution

It introduces new Bloch type theorems for quaternionic slice regular functions, expanding understanding of their geometric and functional properties.

Findings

Proves injective and covering properties of slice regular functions.

Shows existence of a universal ball in the image of certain slice regular functions.

Analyzes properties of functions from slice regular Bloch and Bergman spaces.

Abstract

In this paper we prove two Bloch type theorems for quaternionic slice regular functions. We first discuss the injective and covering properties of some classes of slice regular functions from slice regular Bloch spaces and slice regular Bergman spaces, respectively. And then we show that there exits a universal ball contained in the image of the open unit ball $\mathbb{B}$ in quaternions $\mathbb{H}$ through the slice regular rotation $\widetilde{f}_{u}$ of each slice regular function $f:\overline{\mathbb{B}}\rightarrow \mathbb{H}$ with $f'(0)=1$ for some $u\in \partial\mathbb{B}$.