Geodesic
On a theorem of Lindelof
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 2.

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We give a quasiconformal version of the proof for the classical Lindelof theorem: Let f map the unit disk 𝔻 conformally onto the inner domain of a Jordan curve 𝒞: Then 𝒞 is smooth if and only if arg f'(z) has a continuous extension to 𝔻. Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.
Keywords: Lindelof theorem, infinitesimal geometry, continuous extension to the boundary, differentiability at the boundary, conformal and quaisconformal mappings 
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Gutlyanskii, Vladimir Ya.; Martio, Olli; Ryazanov, Vladimir. On a theorem of Lindelof. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 65 (2011) no. 2. http://geodesic.mathdoc.fr/item/AUM_2011_65_2_a10/

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