Geodesic
Extremal problems in the function theory associated with the $n$-fold symmetry
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 83-111 
V. N. Dubinin; E. V. Kostyuchenko. Extremal problems in the function theory associated with the $n$-fold symmetry. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 17, Tome 276 (2001), pp. 83-111. http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a4/
@article{ZNSL_2001_276_a4,
     author = {V. N. Dubinin and E. V. Kostyuchenko},
     title = {Extremal problems in the function theory associated with the $n$-fold symmetry},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {83--111},
     year = {2001},
     volume = {276},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_276_a4/}
}
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We consider some conventional problems of the theory of functions of a complex variable such that their extremal configurations have the $n$-fold symmetry. We discuss two-point distortion theorems corresponding to the two-fold symmetry. New estimates are obtained for the module of a doubly connected domain. These estimates generalize known results by Rengel, Grötzsch, and Teichmüller to the case of rings with the $n$-fold symmetry, where $n\ge2$. New distortion theorems are proved for functions meromorphic and univalent in a disk or in a ring. In these theorems, the extremal function also has the corresponding symmetry. All of the problems mentioned above are unified by the method applied; this method is based on properties of the conformal capacity and on symmetrization.