Univalence results for slice regular functions of a quaternion variable

Abstract

The goal of this paper is study the univalence property of slice regular functions of quaternion variable. We prove a number of results, among which several sufficient conditions for univalence, an area-type Theorem, and a Bieberbach–de Branges Theorem for a subclass of slice regular functions. We also discuss some geometric and algebraic interpretations of our results in terms of maps from to itself.