An analog of the Schwarz lemma for locally quasiconformal automorphisms of the unit disk

We obtain sharp estimates for the module of functions in the classes of normalized locally quasiconformal authomorphisms of the unit disk with given majorants of M. A. Lavrent’ev’s characteristic. The estimates are analogs of Schwarz’s lemma and A. Mori’s theoremand they imply the classical growth theorems for quasiconformal authomorphisms of the disk. In the classes we also prove sharp estimates of the conformal radius and the radius of covering disk. The main results are obtained by methods of extremal lengths and symmetrization.