Some properties for quaternionic slice-regular functions on domains without real points

TL;DR

This paper extends the theory of quaternionic slice-regular functions to domains without real points, introducing stem functions and generalizing key principles like identity, maximum modulus, and open mapping theorems.

Contribution

It introduces the class of slice regular functions induced by stem functions on non-real domains and extends fundamental properties to this broader setting.

Findings

Extended identity principle for non-real domains

Generalized maximum and minimum modulus principles

Open mapping theorem for slice regular functions without real points

Abstract

The theory of slice regular functions over the quaternions, introduced by Gentili and Struppa in [5], was born on domains that intersect the real axis. This hypothesis can be overcome using the theory of stem functions introduced by Ghiloni and Perotti ([6]), in the context of real alternative algebras. In this paper I will recall the notion and the main properties of stem functions. After that I will introduce the class of slice regular functions induced by stem functions and, in this set, I will extend the identity principle, the maximum and minimum modulus principles and the open mapping theorem. Differences will be shown between the case when the domain does or does not intersect the real axis.