On the uniqueness theorems for the transformations of sets and condensers in the plane

The contraction transformation of the compact sets and the separating transformation of the sets and the condensers in the extended complex plane are considered. The first transformation is well-known in the theory of functions and in the potential theory. The second transformation was introduced by the first-named author and it has many applications in the geometric theory of functions of a complex variable. In the present paper, the necessary and sufficient condition for the conservation of the logarithmic capacity under the contraction transformation is proved. Also, we give the such conditions for the separating transformation of the condensers and domains. As the applications, the uniqueness of the extremal compact set and the extremal configuration in the some known problems of the potential theory is obtained. Our results supplement the uniqueness theorems for the symmetrization transformations which were proved by J. A. Jenkins, I. P. Mityuk, V. A. Shlyk and other mathematicians.