On analogue of Koebe-Bloch theorem for ring homeomorphisms

The paper proves an analogue of the Koebe–Bloch theorem for ring Q-homeomorphisms in Rn (n ≥ 2). The authors consider the family FδK,Q(D) of ring Q-homeomorphisms for which the chordal distance between the image of a compact set K and the boundary of the image of the domain is at least δ > 0. Under the condition that for every point x0 ∈ D and any 0 < r1 < r2 < r0 there exists a subset E1 ⊂ [r1, r2] of positive linear Lebesgue measure on which Q is integrable with respect to the (n−1)-dimensional Hausdorff measure on the spheres S(x0, r), it is shown that the family FδK,Q(D) is uniformly open on K. This means that for any ε0 > 0 there exists r0 > 0 such that the chordal ball Bh(f(x0), r0) is contained in f(B(x0, ε0)) for all f ∈ FδK,Q(D). A special case concerns Orlicz–Sobolev classes W1,φloc under suitable conditions on the outer dilatation KO(x, f). The result generalizes the classical Koebe–Bloch theorem to ring Q-homeomorphisms and is important for the theory of moduli and quasiconformal mappings.