Boundary Julia theory for slice regular functions

The theory of slice regular functions is nowadays widely studied and has found its elegant applications to a 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 the boundary Julia theory for quaternions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Carath\'{e}odory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball \(\mathbb B\subset \mathbb H\) and of the right half-space \(\mathbb H_+\). Especially, we find a new phenomenon that the classical Hopf lemma about \(f'(\xi)>1\) at the boundary point may fail in general in quaternions, and its quaternionic variant should involve the Lie bracket reflecting the non-commutative feature of quaternions.