Loewner-Kufarev equation and extremal problems for conformal mappings with fixed points on the boundary

The report will be devoted to results of the investigation which relates to semigroup aspect of the Loewner’s theory and connected to conformal mappings of the unit disk with the given fixed points on the boundary. There are studied the evolutionary equation in the distinguished semigroup of conformal mappings and a parametric representation of the corresponding class of univalent functions. Some interior problems of the theory of univalent functions with restrictions on the angular derivative in a boundary point are solved. In particular, an analog of the theorem of rotation of Goluzin is obtained. The base of the method and the main terminology have been put in the articles “Semigroups of conformal mappings”, Mat. Sb. (N.S.), 129(171):4 (1986), 451–472, “Fractional iterates of functions analytic in the unit disk, with given fixed points”, Mat. Sb., 182:9 (1991), 1281–1299, which were reported at A. Gonchar’s seminar.