Geodesic
Extremal properties of mappings onto surfaces with parallel slits
Izvestiya. Mathematics, Tome 13 (1979) no. 1, pp. 1-8 
Yu. E. Alenitsyn. Extremal properties of mappings onto surfaces with parallel slits. Izvestiya. Mathematics, Tome 13 (1979) no. 1, pp. 1-8. http://geodesic.mathdoc.fr/item/IM2_1979_13_1_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper an are a theorem is established for certain regular functions associated with multivalent mappings of a finitely connected domain onto a surface with parallel slits. Several consequences of this theorem generalize well-known results from the theory of univalent conformal mappings. The notion of the generalized span of a domain is introduced. It is then shown that it possesses certain properties completely analogous to the basic extremal properties of the span of a domain. Grötzsch's theorem concerning the range of the first coefficient of the regular part of the normalized Laurent expansion of a univalent function about a pole is extended to multivalent functions. Bibliography: 7 titles.

[1] Alenitsyn Yu. E., “Konformnye otobrazheniya mnogosvyaznoi oblasti na mnogolistnye kanonicheskie poverkhnosti”, Izv. AN SSSR. Ser. matem., 28 (1964), 607–644 | MR | Zbl

[2] Alenitsyn Yu. E., “Teoremy ploschadei dlya funktsii, analiticheskikh v konechnosvyaznoi oblasti”, Izv. AN SSSR. Ser. matem., 37 (1973), 1132–1154 | Zbl

[3] Garabedian P. R., Schiffer M., “Identities in the theorie of conformal mapping”, Trans. Amer. Math. Soc., 65:2 (1949), 187–238 | DOI | MR | Zbl

[4] Schiffer M., “The kernel function of an orthonormal system”, Duke Math. J., 13 (1946), 520–540 | DOI | MR

[5] Schiffer M., “The span of multiply connected domains”, Duke Math. J., 10 (1943), 209–216 | DOI | MR | Zbl

[6] Nevanyainna R., Uniformizatsiya, IL, M., 1955

[7] Grötzsch H., “Über das Parallelschlitztheorem der konformen Abbildung schlichter Bereiche”, Ber. Verh. sachs. Akad. Wiss. Leipzig, Math.-phys. Kl, 84, 1932, 15–36 | Zbl