On some questions connected with the problem of existence of automorphic analytic functions with given modulus of boundary values

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.
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