Dvorak I. The method of symmetrization in problems of extremal decomposition of a complex plane

In the dissertation work three problems were investigated, two of which were put in "Uspekhi Matematicheskikh Nauk" in 1994 by V. M. Dubinin as open problems. In the first section of the dissertation the review of the literature is made, The basic ideas of the methods of studying these problems are described, the definitions and the theorems necessary for the formulation and proof of the basic dissertation results. The second section solves the problem of finding the maximum product of internal radii of partially non-intersecting regions with respect to n-radial systems of points on a certain positive degree of γ of the internal radii of partially intersecting regions relative to the origin of the coordinates and the infinitely distant point for much wider intervals parameter γ. In the third section, the problem of finding the maximum product of internal radii of mutually non-cross regions over the points of a single circle for a certain positive degree γof the interior radius of a certain region relative to the origin of the coordinates for =n0,45, n12 and =n0,5, n126. In the fourth section we obtain a solution to the problem of the maximum product of internal radii of mutually non-intersecting regions with an additional condition of symmetry determined by a certain region, which belongs to the origin of coordinates for some positive degree γ of the interior radius of this domain for γ∈ (0;32], n≥ 9.