Julia theory for slice regular functions

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-Carathéodory 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.

To appear in Transactions of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1412.4207

References in corpus (6)

• Twistor transforms of quaternionic functions and orthogonal complex structures
• The Schwarz-Pick lemma for slice regular functions
• A Bloch-Landau Theorem for slice regular functions
• Landau-Toeplitz theorems for slice regular functions over quaternions
• The Growth and Distortion Theorems for Slice Monogenic Functions
• Extremal functions of boundary Schwarz lemma