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On Symmetric Quasicircles

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Shengjian Wu  and
Shanshuang Yang
Shengjian Wu
Affiliation:
Department of Mathematics Peking UniversityBeijing 100871China e-mail: wusj@pku.edu.cn
Shanshuang Yang
Affiliation:
Department of Mathematics and Computer Science Emory UniversityAtlanta GA 30322USA e-mail: syang@mathcs.emory.edu
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Abstract

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We study an important subclass of quasicircles, namely, symmetric quasicircles. Several characterizations for quasicircles, such as the reverse triangle inequality, the M -condition and the quasiconformal extension property, have been extended to symmetric quasicircles by Becker and Pommerenke and by Gardiner and Sullivan. In this paper we establish several relations among various domain constants such as quasiextremal distance constants, (local) reflection constants and (local) extension constants for this class. We also give several characterizations for symmetric quasicircles such as the strong quadrilateral inequality and the strong extremal distance property. They correspond to the quadrilateral inequality and the extremal distance property for quasicircles.

Keywords

quasicirclesymmetricquasisymmetricquasiextremal distancequasiconformalreflectionextensionmodulusdilatation

MSC classification

Secondary: 30C62: Quasiconformal mappings in the plane
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 1 , February 2000 , pp. 131 - 144
Copyright
Copyright © Australian Mathematical Society 2000

References

[Ba]Bagby, T., ‘The modulus of a plane condenser’, J. Math. Mech. 17(1967), 315–329.Google Scholar
[BP]Becker, J. and Pommerenke, C., ‘liber die quasikonforme Fortsetzung schlichter Funktionen’,Math. Z 161 (1978), 69–80.CrossRefGoogle Scholar
[Fe]Fehlmann, R., ‘Über extremale quasikonforme Abbidungen’, Comment. Math. Helv. 56 (1981),558–580.CrossRefGoogle Scholar
[Ga]Gardiner, F. P., Teichmiiller theory and quadratic differentials (Wiley, New York, 1987).Google Scholar
[GS]Gardiner, F. P. and Sullivan, D. P., ‘Symmetric structures on a closed curve’, Amer. J. Math. 114 (1992), 683–736.CrossRefGoogle Scholar
[GY]Garnett, J. B. and Yang, S., ‘Quasiextremal distance domains and integrability of derivatives of conformal mappings’, Michigan Math. J. 41 (1994), 389–406.CrossRefGoogle Scholar
[Ge]Gehring, F. W., ‘Characterizations of quasidisks’, preprint (1997).Google Scholar
[GM]Gehring, F. W. and Martio, O., ‘Quasiextremal distance domains and extension of quasiconformal mappings’, J. Anal. Math. 45 (1985), 181–206.CrossRefGoogle Scholar
[He]Hesse, J., ‘A p-extremal length and p-capacity equality’, Ark. Mat. 13 (1975), 131–144.CrossRefGoogle Scholar
[Ku]Kühnau, R., ‘Moglichst konforme Spiegelung an einer Jordankurve’, Jber. d. Dt. Math. Verein. 90(1988),90–109.Google Scholar
[LV]Lehto, O. and Virtanen, K. I., Quasiconformal mappings in the plane (Springer, New York, 1973).CrossRefGoogle Scholar
[Po]Pommerenke, C., Boundary behaviour of conformal maps (Springer, Berlin, 1992).CrossRefGoogle Scholar
[Wu]Wu, S., ‘Moduli of quadrilaterals and extremal quasiconformal extensions of quasisymmetric functions’, Comment. Math. Helv. 72 (1997), 593–604.CrossRefGoogle Scholar
[Yl]Yang, S., ‘QED domains and NED sets in Rn’ Trans. Amer. Math. Soc. 334 (1992), 97–120.Google Scholar
[Y2]Yang, S., ‘On dilatations and substantial boundary points of homeomorphisms of Jordan curves’, Result. Math. 31 (1997), 180–188.CrossRefGoogle Scholar
[Y3]Yang, S., ‘A modulus inequality for condensers and conformal invariants of smooth Jordan domains’, J. Anal. Math. 75 (1998), 173–183.CrossRefGoogle Scholar