On a~variational problem of Chebotarev in the theory of capacity of plane sets and covering theorems for univalent conformal mappings

S. I. Fedorov

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 115-133

Résumé

This article is devoted to extremal problems in the theory of univalent conformal mappings, related to the moduli of families of curves. In § 1, the problem of finding the minimum capacity in the family of all continua on $\mathbf C$ which contain a fixed quadruple of points which are symmetrically placed with respect to the real axis is solved. Let $R(B,c)$ be the conformal radius of the simply connected region $B$ with respect to the point $c\in B$. In § 2, the maximum of the product $R(B_1,0)R^{-1}(B_2,\infty)$ in the family $\mathscr B(0,\infty;a)$ of all pairs of nonoverlapping simply connected regions $\{B_1,B_2\}$, $0\in B_1$, $\infty\in B_2$, on $\mathbf C\setminus\{a,\overline a,1/a,1/\overline a\}$ is found. Several covering theorems in classes of univalent functions are established as consequences in § 3. Bibliography: 7 titles.