On Koebe-Bloch theorem for ring mappings and convergence to a homeomorphism

A family of mappings is called uniformly light if the image of the continuum under these mappings cannot be contracted to a point under the sequence of mappings of the family. In this paper, we are interested in the problem of the uniform lightness of a family of homeomorphisms satisfying upper moduli inequalities. We have shown that a family of such homeomorphisms satisfies the above-mentioned condition of uniform lightness if the majorant in the corresponding modulus estimate defining the family is integrable over almost all spheres. Under the same conditions, we show that this family of homeomorphisms is uniformly open, i.e. their image contains a ball of fixed radius, independent of each mapping separately. This result generalizes the well-known Koebe-Bloch theorem on the radius of a ball. As an application of the results obtained, we have proved the assertion about the uniform convergence of homeomorphisms to a homeomorphism. As a consequence, we obtained the corresponding results for the Orlicz-Sobolev classes, as well as homeomorphisms inverse to them.