2014/12/13 by Guangbin Ren, Ren, Guangbin, Xieping Wang +1 · 2 citations
Mathematics · #30G35 #32A26 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #math.CV #msc:30G35 #msc:32A26
paper · pdf · doi:10.48550/arxiv.1412.4207
This paper has been rewritten and retitled as "Julia theory for slice regular functions"(see arXiv:1502.02368)
openalex publication_date 2014/12/13 · arxiv created 2020/02/04 · arxiv updated 2020/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-Carathéodory 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⊂ \mathbb H and of the right half-space \mathbb H+. Especially, we find a new phenomenon that the classical Hopf lemma about f'(ξ)>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.