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

Abstract

The theory of slice regular functions over the quaternions, introduced by Gentili and Struppa in 2007, was born on balls centred in the origin and has been extended to more general domains that intersect the real axis in a work of 2009 in collaboration with Colombo and Sabadini. This hypothesis can be overcome using the theory of stem functions introduced by Ghiloni and Perotti in 2011, 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.

Keywords:

• function of a hypercomplex variable
• slice regular functions
• identity principle
• minimum modulus principle
• open mapping theorem

AMS Subject Classifications:

• 30G35
• 30C80

Notes

1 See Remark 2.2