Exact relations are established for the reduced modules of the families of curves separating certain connected subsets in arbitrary connected domains. New explicit expressions for the harmonic measure of a connected boundary subset are obtained as corollaries.

A Jordan curveL is studied, which satisfies an interior or exterior wedge condition for wedges of various geometric forms. We obtain estimates of the continuity moduli for the conformal mappings of the exterior (interior) ofL onto the exterior (interior) of the unit disk.

This article establishes direct and inverse theorems of approximation theory (of the same type as theorems of Dzyadyk) that describe the quantitative connection between the smoothness properties of solutions of the equation , , and the rate of their approximation by module polynomials of the form In particular, a constructive characterization is obtained for generalized Holder classes of such functions on domains with quasiconformal boundary.Bibliography: 19 titles.

This paper is devoted to an application of the methods of geometric function theory of a single complex variable and some results from the theory of conformal and quasiconformal mappings to a solution of the direct problem of the constructive theory of functions. Direct theorems are obtained for the theory of approximation of functions regular in regions bounded by quasiconformal curves admitting a geometric description. In particular, direct theorems are obtained for arbitrary bounded convex regions.Bibliography: 11 items.