On uniformly lightness of one class of mappings and Koebe-Bloch theorem

Abstract

We consider mappings satisfying a certain estimate of the distortion of the modulus of families of paths, similar to the geometric definition of quasiconformal mappings. Under appropriate restrictions, we show that the class of such mappings is uniformly light, i.e., the chordal diameter of the image of continua whose diameter is bounded below is also bounded below uniformly over the class. Under some even greater restrictions, we establish some more explicit estimates of the distortion of the diameters of these continua.