Abstract
Section 1 of the paper is devoted to extremal problems in the classes of conformal homeomorphisms of the circle and the annulus, connected directly with the problem on the maximum of the conformal modulus in the family of doubly connected domains. In Secs. 2 and 3 one considers the class R of functions f(ζ)=c1ζ+c2ζ2+... regular and univalent in the circleU={|ς|<1} and such that f(ζ1)f(ζ2)=1 for ς1ς2∈U (the class of Bieberbach-Eilenberg functions). Here one solves the problem of the maximum of |f′(ς0)| in the class of functions f(ζ)∃R with a fixed value f(ζ0, where ζ0 is an arbitrary point U, and of the maximum of |f′(ς0)| in the entire class R. For the proof one makes use of the method of the moduli of families of curves.
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Literature cited
G. V. Kuz'mina, “Moduli of families of curves and quadratic differntials,” Trudy Mat. Inst. Akad. Nauk SSSR,139 (1980).
G. V. Kuz'mina, “The conformal modulus in families of doubly connected domains that do not contain a given system of points,” J. Sov. Math.,2, No. 6 (1974).
N. A. Lebedev, “On the range of a certain functional in the problem of nonoverlapping domains,” Dokl. Akad. Nauk SSSR,115, No. 6, 1070–1073 (1957).
V. A. Andreev, “Certain problems of nonoverlapping domains,” Sib. Mat. Zh.,17, No. 3, 483–498 (1976).
J. A. Jenkins, “On Bieberbach-Eilenberg functions,” Trans. Am. Math. Soc.,76, No. 3, 389–396 (1954).
J. A. Jenkins, “A remark on “pairs” of regular functions,” Proc. Am. Math. Soc.,31, No. 1, 119–121 (1972).
J. A. Jenkins, On univalent functions omitting two values,” Complex Variables Theory Appl.,3, No. 1–3, 169–172 (1984).
J. A. Hummmel and M. M. Schiffer, “Variational methods for Bieberbach-Eilenberg functions and for pairs,” Ann. Acad. Sci. Fen., Ser. A. I. Math.,3, 3–42 (1977).
Z. Lewandowski, R. Libera, and E. Zlotkiewicz, “Values assumed by Gel'fer functions,” Ann. Univ. Mariae Curie-Sklodowska Sect. A, Math.,31, 75–84 (1977).
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Am. Math. Soc., Providence (1969).
P. Duren and G. Schober, “Nonvanishing univalent functions,” Math. Z.,170, No. 3, 195–216 (1980).
D. Hamilton, “Extremal problems for nonvanishing univalent functions,” in: Aspects of Contemporary Complex Analysis (Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979).
P. M. Tamrazov, “Extremal conformal mappings and poles of quadratic differentials,” Izv. Akad. Nauk SSSR, Ser. Mat.,32, No. 5, 1033–1043 (1968).
G. V. Kuz'mina, On a certain modulus problem for families of curves.” Preprint LOMI, R-6-83, Leningrad (1983).
E. G. Emel'yanov, “Some properties of the moduli of families of curves,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,144, 72–82 (1985).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 94–114, 1985.
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Emel'yanov, E.G., Kuz'mina, G.V. Extremal problems in classes of univalent functions, omitting prescribed values. J Math Sci 38, 2098–2114 (1987). https://doi.org/10.1007/BF01474444
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DOI: https://doi.org/10.1007/BF01474444