doi: 10.2478/v10062-011-0012-7 ANNALES UNIVERSITATIS MARIAE CURIE-SKLODOWSKA LUBLIN – POLONIA VOL. LXV, NO. 2, 2011 SECTIO A 45–51 VLADIMIR GUTLYANSKI ˘ I, OLLI MARTIO and VLADIMIR RYAZANOV On a theorem of Lindel¨of Dedicated to the memory of Professor Jan G. Krzyż Abstract. We give a quasiconformal version of the proof for the classical Lindel¨oftheorem: 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. 1. Introduction. Let f : D → C be a conformal mapping of the unit disk D onto f (D). The smoothness of ∂f (D) yields the smoothness of f on ∂ D. The classical Lindel¨of theorem [7] as well as Warschawski’s theorem [9] on differentiability of f at the boundary ∂ D are the basic results of this kind of behavior. In this paper we adopt a different point of view. Assuming that the boundary curve is smooth, i.e. it has a continuously turning tangent, we extend f over the unit disk to a quasiconformal mapping and apply some results from the infinitesimal geometry of quasiconformal mappings devel- oped in [5], see also [4]. In order to illustrate our approach, we give a quasiconformal version of the proof for the aforementioned Lindel¨of theo- rem. Recall that the standard proof of the Lindel¨ of theorem is based on the 2000 Mathematics Subject Classification. 30C55, 30C60. Key words and phrases. Lindel¨of theorem, infinitesimal geometry, continuous exten- sion to the boundary, differentiability at the boundary, conformal and quaisconformal mappings.