The boundary behavior of conformal mappings with quasiconformal extensions

The boundary behavior of conformal mappings with quasiconformal extensions, were studied. It was assumed that a smooth closed Jordan curve is quasicircle, and the conformal mapping can be extended to a quasiconformal mapping on the complex plane. The image of the open arc of the unit circle under the conformal mapping is found to be a quasiconformal Jordan curve, which can have very complicated geometric structure. Meanwhile, the dependence of the analytical and geometric properties of a conformal mapping in a neighborhood of a boundary point on the degree of regularity of the quasiconformal extension was also analyzed.