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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)
##### MSC:
 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
quasiconformal curve; asymptotically conformal curve