Condenser capacity and symmetrization in the extremal decomposition problems

Capacity approach and Steiner symmetrization are applied to the solution of two extremal decomposition problems on the Riemann sphere. First, the inequality between generalized reduced moduli with respect to the inner points and the reduced moduli of the strips and half–strips are established. The second problem comes under the heading of extremal decompositions with the free poles on the concentric circles. The results obtained are complementary to some classical and recent theorems in various ways.