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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Two-point distortion theorems for harmonic and pluriharmonic mappings


Authors: Peter Duren, Hidetaka Hamada and Gabriela Kohr
Journal: Trans. Amer. Math. Soc. 363 (2011), 6197-6218
MSC (2010): Primary 32H02; Secondary 30C45
Published electronically: July 26, 2011
MathSciNet review: 2833550
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Abstract: Two-point distortion theorems are obtained for affine and linearly invariant families of harmonic mappings on the unit disk, with generalizations to pluriharmonic mappings of the unit ball in $ {\mathbb{C}}^{n}$. In particular, necessary and sufficient conditions are given for a locally univalent harmonic or pluriharmonic mapping to be univalent. Some particular subclasses are also considered.


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Additional Information

Peter Duren
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109–1043
Email: duren@umich.edu

Hidetaka Hamada
Affiliation: Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku Fukuoka 813-8503, Japan
Email: h.hamada@ip.kyusan-u.ac.jp

Gabriela Kohr
Affiliation: Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogăl- niceanu Str., 400084 Cluj-Napoca, Romania
Email: gkohr@math.ubbcluj.ro

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05596-0
PII: S 0002-9947(2011)05596-0
Keywords: Harmonic mapping, pluriharmonic mapping, two-point distortion, affine invariance, linear invariance, univalence, convex mapping, close-to-convex mapping, starlike mapping
Received by editor(s): August 7, 2009
Published electronically: July 26, 2011
Additional Notes: The second author was partially supported by Grant-in-Aid for Scientific Research (C) No. 22540213 from Japan Society for the Promotion of Science, 2011.
The third author was supported by the UEFISCSU-CNCSIS Grant PN-II-ID 524/2007.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.