SUFFICIENT AND NECESSARY CONDITIONS FORCONFORMALITY. PART II. ANALYTIC VIEWPOINT

摘要

This is the second of two papers devoted to the topic of conformality at a point and related notions in the plane. We derive representation formulas and estimates for the mod- ules of families of curves that are images of circles, radial segments and arcs of spirals under a pt-homeomorphism. We use them to convert the extremal-length type sufficient and necessary con- ditions for conformality at a point from Part I to analytic sufficient conditions, that depend on directional dilatations and bypass the assumption of K-quaisconformality. Our results extend the Teichmiiller—Wittich—Belinskii theorem, results of Reich and Walczak, the author and Jenkins, and Gutlyanskii and Martio.