Geodesic
On some questions connected with the problem of existence of automorphic analytic functions with given modulus of boundary values
Sbornik. Mathematics, Tome 39 (1981) no. 4, pp. 501-518 Cet article a éte moissonné depuis la source Math-Net.Ru

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For an arbitrary region $D$ in the extended plane it is shown that the possibility of constructing automorphic analytic functions with given modulus of boundary values on the universal covering surface of $D$ is a local property of the boundary of $D$ and is equivalent to the possibility of giving a good description of the extremals for a rather large class of problems in this region. Bibliography: 12 titles. 
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M. V. Samokhin. On some questions connected with the problem of existence of automorphic analytic functions with given modulus of boundary values. Sbornik. Mathematics, Tome 39 (1981) no. 4, pp. 501-518. http://geodesic.mathdoc.fr/item/SM_1981_39_4_a4/

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[8] W. Rudin, “Some theorems on bounded analityc functions”, Trans. Amer. Math. Soc., 78 (1955), 333–342 | DOI | MR | Zbl

[9] T. Gamelin, “Localisation of corona problem”, Pacific J. Math., 34 (1970), 73–81 | MR | Zbl

[10] T. Gamelin, “Extremal problems in arbitrary domains”, Michigan Math. J., 20 (1973), 3–11 | DOI | MR | Zbl

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[12] S. Ya. Khavinson, “O probleme Karateodori–Feiera dlya analiticheskikh funktsii v konechno-svyaznykh oblastyakh”, Uchenye zapiski Vladimirskogo gos. ped. in-ta fiz.-matem. seriya, vyp. II, 1955