Conformal invariant functionals on the Riemann sphere

The main aim of this work is to establish new inequalities for the Grunsky coefficients of univalent functions. For this purpose, we apply results from the theory of problems on extremal decomposition. To obtain inequalities for the Grunsky coefficients of a function $f\in\Sigma$, we apply a solution of the problem on the maximum of a conformal invariant (this invariant, in its turn, is connected with the problem on extremal decomposition of $\overline{\mathbb C}$ into a family of simply connected and doubly connected domains). In contrast to similar inequalities obtained from the Jenkins general coefficient theorem, the inequalities established in this work are valid without any restrictions on the initial coefficients of the expansion of a function $f\in\Sigma$.