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The problem of conformal transformations of a circle into nonoverlapping regions

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Abstract

Let a, a≠0, a≠∞, be a fixed point in the z-plane, ℜ (a, 0, ∞), the class of all systemsf k(ζ)l 3 of functions z=f k(ζ), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦ζ¦<1, and the third maps in a similar manner the region ¦ζ¦>1, into pair-wise nonintersecting regions Bk, k=1, 2, 3, containing the points a, 0, and ∞, respectively, so thatf 1(0)=a,f 2(0)=0 andf 3(∞)=∞. The region of values ℰ (a, 0, ∞) of the system M(¦f 1'(0)¦, ¦f 2'(0)¦, 1/¦f 3'(∞)¦) in the class ℜ(a, 0, ∞) is determined.

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Literature cited

  1. N. A. Lebedev, “On the theory of conformal mappings of a circle into non-overlapping regions,” Dokl. Akad. Nauk SSSR,103, No. 4, 553–555 (1955).

    Google Scholar 

  2. G. M. Goluzin, The Geometrical Theory of Functions of a Complex Variable [in Russian], Moscow (1966).

  3. L. I. Kolbina, “Some extremal problems in conformal mappings,” Dokl. Akad. Nauk SSSR,84, No. 5, 865–868 (1952).

    Google Scholar 

  4. N. A. Lebedev, “On the region of values of a functional in the problem of non-overlapping regions,” Dokl. Akad. Nauk SSSR,115, No. 6, 1070–1073 (1957).

    Google Scholar 

  5. L. I. Kolbina, “The conformal mapping of the unit circle into nonoverlapping regions,” Vestnik Leningr. Un-ta, No. 5, Issue 2, 37–43 (1955).

    Google Scholar 

  6. N. A. Lebedev, “The majorant region for the expression\(I = \ln \frac{{z^\lambda f'(z)^{1 - \lambda } }}{{f(z)^\lambda }}\) in the class S,” Vestnik Leningr. Un-ta, No. 8, Issue 3, 29–41 (1955).

    Google Scholar 

  7. V. V. Golubev, Lectures on the Analytical Theory of Differential Equations [in Russian], Moscow-Leningrad (1950).

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Authors and Affiliations

  1. A. A. Zhdanov Leningrad State University, USSR

    L. Kh. Burshtein

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  1. L. Kh. Burshtein
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Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.

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Burshtein, L.K. The problem of conformal transformations of a circle into nonoverlapping regions. Mathematical Notes of the Academy of Sciences of the USSR 6, 705–709 (1969). https://doi.org/10.1007/BF01093806

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  • DOI: https://doi.org/10.1007/BF01093806

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