Julia theory for slice regular functions

Abstract

The theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of Julia theory for slice regular functions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Caratheodory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball B ⊂ H and of the right half-space H+. We provide an explicit example to show that in quaternionic Hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real, in contrast to the complex version. Our result implies that the commonly believed version of the Hopf lemma turns out to be totally wrong. This new quaternionic version of the Hopf lemma also improves Osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis, since the result involves the Lie brackets which reflect the non-commutative feature of quaternions.