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On the rectifiability and smoothness of some quasiconformal curves. (English. Russian original) Zbl 0861.30020
Russ. Acad. Sci., Dokl., Math. 50, No. 2, 292-294 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 338, No. 5, 583-584 (1994).
In this paper the authors give sufficient conditions for the rectifiability and the smoothness of a quasiconformal curve \(\Gamma\) which are based on properties of the complex characteristic \(\mu = \mu(z)\) belonging to a quasiconformal continuation of a conformal mapping from the unit disc onto the interior of \(\Gamma\). Assuming the convergence of \[ \iint_{1 < |z |< R} {\bigl|\mu(z) \bigr|^\alpha \over \bigl(1- |z |^2} dm_z \] for some \(\alpha \in(0,2]\) they prove that \(\Gamma\) has finite length. In the case \(\alpha \in (0,1]\), the authors show the smoothness of \(\Gamma\). Unfortunately, there are a number of misprints, for instance in the main condition (3) and in Theorem 1, where in the English translation \(\ln f(z)\) should be read as \(\ln f'(z)\).
Reviewer: E.Hoy (Friedberg)
30C62 Quasiconformal mappings in the complex plane