Abstract:
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.
Fulltext:
PDF file (1054 kB)
