Julia theory for slice regular functions

TL;DR

This paper develops a quaternionic Julia theory for slice regular functions, establishing key boundary theorems and revealing differences from complex analysis due to quaternion non-commutativity.

Contribution

It introduces quaternionic versions of fundamental Julia theorems for slice regular functions, highlighting novel boundary behaviors and correcting misconceptions from complex analysis.

Findings

Quaternionic Julia lemmas established

Boundary Schwarz lemma involves Lie brackets

Slice derivatives at boundary points can be non-positive

Abstract

Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball $\mathbb B$ and of the right half-space $\mathbb H^+$. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quaternions. Together with some explicit examples, it shows that the slice derivative of a slice regular self-mapping of $\mathbb B$ at a boundary fixed point is not necessarily a positive real number, in contrast to that in the complex case, meaning that its commonly believed version turns out to be totally wrong.