On a theorem of Lindelof
Vladimir Ya. Gutlyanskii; Olli Martio; Vladimir Ryazanov
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)
- Volume: 65, Issue: 2
- ISSN: 0365-1029
Access Full Article
top
Abstract
topHow to cite
topVladimir Ya. Gutlyanskii, Olli Martio, and Vladimir Ryazanov. "On a theorem of Lindelof." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.2 (2011): null. <http://eudml.org/doc/289742>.
@article{VladimirYa2011,
abstract = {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 $f^\{\prime \}(z)$ has a continuous extension to $\overline\{\mathbb \{D\}\}$. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.},
author = {Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Lindelof theorem; infinitesimal geometry; continuous extension to the boundary; differentiability at the boundary; conformal and quaisconformal mappings},
language = {eng},
number = {2},
pages = {null},
title = {On a theorem of Lindelof},
url = {http://eudml.org/doc/289742},
volume = {65},
year = {2011},
}
TY - JOUR
AU - Vladimir Ya. Gutlyanskii
AU - Olli Martio
AU - Vladimir Ryazanov
TI - On a theorem of Lindelof
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 2
SP - null
AB - 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 $f^{\prime }(z)$ has a continuous extension to $\overline{\mathbb {D}}$. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
LA - eng
KW - Lindelof theorem; infinitesimal geometry; continuous extension to the boundary; differentiability at the boundary; conformal and quaisconformal mappings
UR - http://eudml.org/doc/289742
ER -
References
top- 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.
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.