Advertisement

Ukrainian Mathematical Journal

, Volume 64, Issue 1, pp 24–34 | Cite as

On one Dubinin extreme problem

  • Ya. V. Zabolotnyi
Article

We obtain a special solution of the well-known Dubinin conjecture on disjoint domains in the complex plane.

Keywords

Conformal Mapping Extreme Problem Imaginary Axis Geometric Theory Quadratic Differential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. A. Lavrent’ev, “On the theory of conformal mappings,” Trudy Fiz.-Mat. Inst. Akad. Nauk SSSR, 5, 159–245 (1934).Google Scholar
  2. 2.
    G. M. Goluzin, Geometric Theory of Functions of Complex Variable [in Russian], Nauka, Moscow (1966).Google Scholar
  3. 3.
    W. K. Hayman, Multivalent Functions, Cambridge University, Cambridge (1958).zbMATHGoogle Scholar
  4. 4.
    J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin (1958).CrossRefGoogle Scholar
  5. 5.
    L. I. Kolbina, “Conformal mapping of a unit disk onto disjoint domains,” Vestn. Leningrad. Univ., 5, 37–43 (1955).MathSciNetGoogle Scholar
  6. 6.
    G. P. Bakhtina, Variational Methods and Quadratic Differentials in the Problems of Disjoint Domains [in Russian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Kiev (1975).Google Scholar
  7. 7.
    V. N. Dubinin, “Method of symmetrization in the geometric theory of functions of complex variable,” Usp. Mat. Nauk, 49, No. 1(295), 3–76 (1994).MathSciNetGoogle Scholar
  8. 8.
    V. N. Dubinin, “Separating transformation of domains and the problems of extreme partition,” Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR, 168, 48–66 (1988).Google Scholar
  9. 9.
    A. K. Bakhtin, G. P. Bakhtina, and Yu. B. Zelinskii, Topological-Algebraic Structures and Geometric Methods in Complex Analysis [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2008).Google Scholar
  10. 10.
    A. K. Bakhtin, “Inequalities for the inner radii of disjoint domains and open sets,” Dop. Nats. Akad. Ukr., No. 10, 7–13 (2006).MathSciNetGoogle Scholar
  11. 11.
    A. K. Bakhtin, “Extreme problems of disjoint domains with free poles on a circle,” Ukr. Mat. Zh., 58, No. 7, 868–886 (2006); English translation: Ukr. Math. J., 58, No. 7, 981–1000 (2006).MathSciNetCrossRefGoogle Scholar
  12. 12.
    A. K. Bakhtin, “Sharp estimates for the inner radii of the systems of disjoint domains and open sets,” Ukr. Mat. Zh., 59, No. 12, 1601–1618 (2007); English translation: Ukr. Math. J., 59, No. 12, 1800–1818 (2007).MathSciNetCrossRefGoogle Scholar
  13. 13.
    O. K. Bakhtin, “Inequalities for the inner radii of the systems of disjoint domains and open sets,” Ukr. Mat. Zh., 61, No. 5, 596–610 (2009); English translation: Ukr. Math. J., 61, No. 5, 716–733 (2009).MathSciNetCrossRefGoogle Scholar
  14. 14.
    B. V. Shabat, Introduction to Complex Analysis [in Russian], Nauka, Moscow (1972).Google Scholar
  15. 15.
    G. V. Kuz’mina, “On the problem of extreme partition of a Riemann sphere,” Zap. Nauch. Sem. LOMI, 185, 96–110 (1990).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2012

Authors and Affiliations

  • Ya. V. Zabolotnyi
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

Personalised recommendations