Emel’yanov, E. G. Conformally invariant functionals on the Riemann sphere. (English. Russian original) Zbl 1071.30014 J. Math. Sci., New York 118, No. 1, 4808-4821 (2003); translation from Zap. Nauchn. Semin. POMI 276, 134-154 (2001). Summary: 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 \(f \in \Sigma\). Bibliography: 6 titles. Cited in 1 Document MSC: 30C50 Coefficient problems for univalent and multivalent functions of one complex variable 30C75 Extremal problems for conformal and quasiconformal mappings, other methods 30F15 Harmonic functions on Riemann surfaces × Cite Format Result Cite Review PDF Full Text: DOI