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We give a quasiconformal version of the proof for the classical Lindelof theorem: Let \(f\) map the unit disk \(\mathbb{D}\) conformally onto the inner domain of a Jordan curve \(\mathcal{C}\): Then \(\mathcal{C}\) is smooth if and only if arg \(f'(z)\) has a continuous extension to \(\overline{\mathbb{D}}\). Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
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Tom
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wydano
2011
online
2016-07-27
Twórcy
Bibliografia
- Ahlfors, L. V., Quasiconformal reflections, Acta Math. 109 (1963), 291-301.
- Ahlfors, L. V., Lectures on Quasiconformal Mappings, D. Van Nostrand Co., Inc., Toronto, Ont., 1966; Reprinted by Wadsworth & Brooks, Monterey, CA, 1987.
- Gutlyanskii, V. Ya., Ryazanov, V. I., On asymptotically conformal curves, Complex Variables Theory Appl. 25 (1994), 357-366.
- Gutlyanskii, V. Ya., Ryazanov, V. I., On the theory of the local behavior of quasiconformal mappings, Izv. Math. 59 (1995), no. 3, 471-498.
- Gutlyanskii, V. Ya., Martio, O., Ryazanov, V. I. and Vuorinen, M., Infinitesimal geometry of quasiregular mappings, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 1,
- 101-130.
- Lehto, O., Virtanen, K. I., Quasiconformal Mappings in the Plane, 2nd Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1973.
- Lindelof, E., Sur la representation conforme d’une aire simplement connexe sur l’aire d’un cercle, Quatrieme Congres des Mathematiciens Scandinaves, Stockholm, 1916, pp. 59-90.
- Pommerenke, Ch., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin-Heidelberg-New York, 1992.
- Warschawski, S. E., On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614-620.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_45-51