On a theorem of Lindelof

Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov

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 \(f'(z)\) has a continuous extension to \(\overline{\mathbb{D}}\). Our proof does not use the Poisson integral representation of harmonic functions in the unit disk.

Keywords


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

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References


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DOI: http://dx.doi.org/10.2478/v10062-011-0012-7
Date of publication: 2016-07-27 21:54:08
Date of submission: 2016-07-26 20:50:55


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Copyright (c) 2011 Vladimir Ya. Gutlyanskii, Olli Martio, Vladimir Ryazanov