Carath\'eodory theorems for Slice Regular Functions

TL;DR

This paper extends classical complex analysis theorems to quaternionic slice regular functions, establishing sharp and equivalent versions of Carathéodory and Borel-Carathéodory theorems for functions with positive real part.

Contribution

It provides the first sharp quaternionic Carathéodory theorem and shows the equivalence with the quaternionic Borel-Carathéodory theorem, relaxing previous restrictions.

Findings

Established a sharp quaternionic Carathéodory theorem.

Proved the equivalence of Carathéodory and Borel-Carathéodory theorems in quaternionic setting.

Extended the class of functions with positive real part for which these theorems hold.

Abstract

In this paper a quaternionic sharp version of the Carath\'{e}odory theorem is established for slice regular functions with positive real part, which strengthes a weaken version recently established by D. Alpay et. al. using the Herglotz integral formula. Moreover, the restriction of positive real part can be relaxed so that the theorem becomes the quaternionic version of the Borel-Carath\'{e}odory theorem. It turns out that the two theorems are equivalent.