Geodesic
Carath\'eodory domains and Rudin's converse of the maximum modulus principle
Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 161-174

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We obtain extensions of the classical Rudin theorem on the converse of the maximum modulus principle from the unit disc to Carathéodory domains. The proofs are based on recent results about properties of conformal mappings of Carathéodory domains, which are also considered in the paper. Bibliography: 18 titles.
Keywords: Carathéodory domain, maximum modulus principle, meromorphic functions, equality in the sense of conformal mappings. 
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     author = {K. Yu. Fedorovskiy},
     title = {Carath\'eodory domains and {Rudin's} converse of the maximum modulus principle},
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K. Yu. Fedorovskiy. Carath\'eodory domains and Rudin's converse of the maximum modulus principle. Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 161-174. http://geodesic.mathdoc.fr/item/SM_2015_206_1_a9/