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

In the classes of normalized locally quasiconformal authomorphisms of the unit disk with the given majorant of M. A. Lavrentev's characteristic we obtain sharp estimates of the module of function as analogs of Schwarz's lemma and A. Mori's theorem. The classical growth theorems for quasiconformal authomorphisms of the disk follow from proved inequalities. In the classes of normalized locally quasiconformal homeomorphisms of the unit disk with the given majorant of M. A. Lavrentev's characteristic we prove sharp estimates of the conformal radius and radius of covering disk. The main results are obtained by the means of the methods of extremal lengths and symmetrization.