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The Schwarz lemma in Clifford analysis


Author: Zhongxiang Zhang
Journal: Proc. Amer. Math. Soc. 142 (2014), 1237-1248
MSC (2010): Primary 30G35
Published electronically: January 6, 2014
MathSciNet review: 3162246
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Abstract: In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.


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Additional Information

Zhongxiang Zhang
Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
Email: zhxzhang.math@whu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-11854-5
Keywords: Clifford algebra, Poisson integral, M\"obius transformation, Schwarz lemma
Received by editor(s): December 19, 2011
Received by editor(s) in revised form: April 29, 2012
Published electronically: January 6, 2014
Additional Notes: The author was supported by the DAAD K. C. Wong Education Foundation and the NNSF for Young Scholars of China (No. 11001206).
Communicated by: Richard Rochberg
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.