On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable

TL;DR

This paper explores geometric properties of slice regular quaternion functions, establishing key theorems like Area-type, Rogosinski, and Bieberbach-de Branges, with interpretations in four-dimensional space.

Contribution

It introduces new geometric concepts for slice regular functions and defines a novel composition method, advancing the understanding of quaternionic function theory.

Findings

Proved an Area-type Theorem for slice regular functions

Established a Rogosinski inequality in this context

Derived a Bieberbach-de Branges Theorem for a subclass

Abstract

The goal of this paper is to introduce and study some geometric properties of slice regular functions of quaternion variable like univalence, subordination, starlikeness, convexity and spirallikeness in the unit ball. We prove a number of results, among which an Area-type Theorem, Rogosinski inequality, 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 $\mathbb R^4$ to itself. As a tool for subordination we define a suitable notion of composition of slice regular functions which is of independent interest.