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Julia theory for slice regular functions


Authors: Guangbin Ren and Xieping Wang
Journal: Trans. Amer. Math. Soc. 369 (2017), 861-885
MSC (2010): Primary 30G35, 30C80, 32A40, 31B25
DOI: https://doi.org/10.1090/tran/6717
Published electronically: March 18, 2016
MathSciNet review: 3572257
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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.


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Additional Information

Guangbin Ren
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
Email: rengb@ustc.edu.cn

Xieping Wang
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
Email: pwx@mail.ustc.edu.cn

DOI: https://doi.org/10.1090/tran/6717
Keywords: Quaternions, slice regular functions, Julia's lemma, Julia-Carath\'eodory theorem, boundary Schwarz lemma, Burns-Krantz rigidity theorem, Hopf's lemma.
Received by editor(s): January 28, 2015
Published electronically: March 18, 2016
Additional Notes: This work was supported by the NNSF of China (11371337), RFDP (20123402110068).
Article copyright: © Copyright 2016 American Mathematical Society