Geodesic
On a variational problem of Chebotarev in the theory of capacity of plane sets and covering theorems for univalent conformal mappings
Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 115-133 
S. I. Fedorov. On a variational problem of Chebotarev in the theory of capacity of plane sets and covering theorems for univalent conformal mappings. Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 115-133. http://geodesic.mathdoc.fr/item/SM_1985_52_1_a6/
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     title = {On a~variational problem of {Chebotarev} in the theory of capacity of plane sets and covering theorems for univalent conformal mappings},
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     url = {http://geodesic.mathdoc.fr/item/SM_1985_52_1_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[2] Kuzmina G. V., Moduli semeistv krivykh i kvadratichnye differentsialy, Trudy MIAN, 139, 1980 | MR | Zbl

[3] Kuzmina G. V., “Teoremy pokrytiya v klassakh funktsii Biberbakha–Eilenberga”, Zapiski nauchn. semin. LOMI, 112, 1981, 143–158 | MR | Zbl

[4] Kuzmina G. V., “K zadache o maksimume proizvedeniya konformnykh radiusov nenalegayuschikh oblastei”, Zapiski nauchn. semin. LOMI, 100, 1980, 131–145 | MR | Zbl

[5] Tamrazov P. M., “Teoremy pokrytiya linii pri konformnom otobrazhenii”, Matem. sb., 66 (108) (1965), 502–524 | MR | Zbl

[6] Dzhenkins Dzh., Odnolistnye funktsii i konformnye otobrazheniya, IL, M., 1962

[7] Jenkins J. A., “On certain geometrical problems associated with capacity”, Math. Nachr. B, 39:4–6 (1969), 349–356 | DOI | MR | Zbl