On a theorem of Lindelof

• Autorzy: Vladimir Ya. Gutlyanskii , Olli Martio , Vladimir Ryazanov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Lindelof theorem; infinitesimal geometry; continuous extension to the boundary; differentiability at the boundary; conformal and quaisconformal mappings

• Wydawca: Wydawnictwo Uniwersytetu Marii Curie-Skłodowskiej
• Czasopismo: Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
• Rocznik: 2011
• Tom: 65
• Numer: 2

Daty

wydano

2011

online

2016-07-27

Twórcy

autor

Vladimir Ya. Gutlyanskii

autor

Olli Martio

autor

Vladimir Ryazanov

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.

Identyfikatory

DOI

10.17951/a.2011.65.2.45-51