Denega I. Quadratic differentials and symmetrization methods in problems on extremal decomposition of the complex plane

The thesis is devoted to the development of new and improvement of existing approaches and methods of research of open extremal problems of geometric function theory of a complex variable. The main object of the study is the extremal problems with fixed and free poles of the corresponding quadratic differentials. In thesis, an effective upper estimates are obtained for the products of inner radii of mutually non-overlapping domains with fixed poles corresponding quadratic differentials on the (n,m)-radial systems of points of the complex plane at all possible values of the degree y є (0, nm] of the inner radius of the domain relative to the origin (the degree y є R+ of the inner radii of the domains relative to the origin and the infinitely distant point). The corresponding results are obtained for the case when the poles corresponding quadratic differentials are located on the unit circle and in the case when the domains are mirror-symmetric relative to the unit circle. The conditions under which in the proved results the structure of points and domains is irrelevant are established. Proved estimates of functionals have made it possible to find some exact solutions in open extremal problems on mutually non-overlapping domains. In particular, an open problem of finding the maximum of product of inner radii of two domains relative to the points of a unit circle on the degree y of the inner radius of the domain relative to the origin at arbitrary y є (0, 2], provided that all three domains are mutually non-overlapping domains is solved. And it is generalized for the case of two arbitrary points on the complex plane. An upper estimates are given for products of inner radii of mutually non-overlapping domains with respect to the points located on one line at all possible values of the degree y of the inner radius of the domain relative to the origin (the degree y of the inner radii of the domains relative to the origin and the infinitely distant point, the degree y of the inner radius of the domain relative to the infinitely distant point). As a consequence, the results are obtained when two rays contain the same number of points. On the basis of the proved estimations it is possible to obtain a number of new estimations for the functions realizing conformal mapping of a circle on domains, with some special properties. The results can be applied to coverage theorems, distortion theorems, estimates of coefficients of univalent functions and in some problems of holomorphic dynamics.