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QUASICONFORMAL EXTENSIONS OF HARMONIC MAPPINGS WITH A COMPLEX PARAMETER

Part of: Geometric function theory Two-dimensional theory

Published online by Cambridge University Press:  19 September 2016

XINGDI CHEN  and
YUQIN QUE
XINGDI CHEN*
Affiliation:
Department of Mathematics, Huaqiao University, Quanzhou, Fujian 362021, PR China email [email protected]
YUQIN QUE
Affiliation:
Department of Mathematics, Huaqiao University, Quanzhou, Fujian 362021PR China email [email protected]
*
✉[email protected]
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Abstract

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In this paper, we study quasiconformal extensions of harmonic mappings. Utilizing a complex parameter, we build a bridge between the quasiconformal extension theorem for locally analytic functions given by Ahlfors [‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud.79 (1974), 23–29] and the one for harmonic mappings recently given by Hernández and Martín [‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math.38 (2) (2013), 617–630]. We also give a quasiconformal extension of a harmonic Teichmüller mapping, whose maximal dilatation estimate is asymptotically sharp.

Keywords

harmonic mappingTeichmüller mappingquasiconformal extensionharmonic quasiconformal mapping

MSC classification

Primary: 30C62: Quasiconformal mappings in the plane 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C55: General theory of univalent and multivalent functions
Secondary: 31A05: Harmonic, subharmonic, superharmonic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 307 - 315
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

This paper is supported by NNSF of China (11471128), the Natural Science Foundation of Fujian Province of China (2014J01013), NCETFJ Fund (2012FJ-NCET-ZR05), Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX110).

References

Ahlfors, L. V., ‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud. 79 (1974), 23–29.Google Scholar
Becker, J., ‘Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte functionen’, J. reine angew. Math. 255 (1972), 23–43.Google Scholar
Becker, J. and Pommerenke, Ch., ‘Schlichtheitskriterier und Jordangebiete’, J. reine angew. Math. 354 (1984), 74–94.Google Scholar
Chen, X. D. and Fang, A. N., ‘Harmonic Teichmüller mappings’, Proc. Japan Acad. Ser. A Math. Sci. 82(7) (2006), 101–105.Google Scholar
Chuaqui, M., Duren, P. and Osgood, B., ‘The Schwarzian derivative for harmonic mappings’, J. Anal. Math. 91 (2003), 329–351.CrossRefGoogle Scholar
Chuaqui, M., Duren, P. and Osgood, B., ‘Ellipses, near ellipses, and harmonic Möbius transformations’, Proc. Amer. Math. Soc. 133 (2005), 2705–2710.Google Scholar
Hernández, R. and Martín, M. J., ‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math. 38(2) (2013), 617–630.CrossRefGoogle Scholar
Hernández, R. and Martín, M. J., ‘Pre-Schwarzian and Schwarzian derivatives of harmonic mappings’, J. Geom. Anal. 25(1) (2015), 64–91.CrossRefGoogle Scholar
Lehto, O., Univalent Functions and Teichmüller Spaces (Springer, New York–Heidelberg, 1987).Google Scholar
Lewy, H., ‘On the non-vanishing of the Jacobian in certain one-to-one mappings’, Bull. Amer. Math. Soc. (N.S.) 42 (1936), 689–692.Google Scholar