2015/02/09 by Guangbin Ren, Ren, Guangbin, Xieping Wang +1 · 1 citation
Mathematics · #Algebraic and Geometric Analysis #Holomorphic and Operator Theory #Mathematics and Applications #math.CV #msc:30C80 #msc:30G35 #msc:31B25 #msc:32A40
paper · pdf · doi:10.48550/arxiv.1502.02368
To appear in Transactions of the American Mathematical Society. arXiv admin note: substantial text overlap with arXiv:1412.4207
arxiv created 2016/03/20 · arxiv updated 2016/03/22
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.