Skip to main content
SpringerLink
Log in

Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. A. Lavrent’ev, “On the theory of conformal mappings,” Tr. Fiz.-Mat. Inst. Akad. Nauk SSSR, 5, 159–245 (1934).

    Google Scholar 

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

    Google Scholar 

  3. W. K. Hayman, Multivalent Functions, Cambridge University Press, Cambridge (1958).

    MATH  Google Scholar 

  4. J. A. Jenkins, Univalent Functions and Conformal Mapping, Springer, Berlin (1958).

    MATH  Google Scholar 

  5. N. A. Lebedev, Area Principle in the Theory of Univalent Functions [in Russian], Nauka, Moscow (1975).

    MATH  Google Scholar 

  6. P. M. Tamrazov, “Extremal conformal mappings and poles of quadratic differentials,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 5, 1033–1043 (1968).

    MATH  MathSciNet  Google Scholar 

  7. G. P. Bakhtina, “On conformal radii of symmetric nonoverlapping domains,” in: Contemporary Problems of Real and Complex Analysis [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1984) pp. 21–27.

    Google Scholar 

  8. G. P. Bakhtina, Variational Methods and Quadratic Differentials in Problems of Nonoverlapping Domains [in Russian], Candidate-Degree Thesis (Physics and Mathematics), Kiev (1975).

  9. G. V. Kuz’mina, Moduli of Families of Curves and Quadratic Differentials [in Russian], Nauka, Leningrad (1980).

    Google Scholar 

  10. G. V. Kuz’mina, “Methods of the geometric theory of functions. I, II,” Algebra Analiz, 9, No. 3, 41–103 (1997); No. 5, 1–50 (1997).

    MathSciNet  Google Scholar 

  11. V. N. Dubinin, “Separating transformation of domains and problems of extremal division,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR, 168, 48–66 (1988).

    Google Scholar 

  12. V. N. Dubinin, “Method of symmetrization in the geometric theory of functions of a complex variable,” Usp. Mat. Nauk, 49, No. 1, 3–76 (1994).

    MathSciNet  Google Scholar 

  13. V. N. Dubinin, “Asymptotics of the modulus of a degenerate condenser and its applications,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR, 237, 56–73 (1997).

    Google Scholar 

  14. V. N. Dubinin, Capacities of Condensers in the Geometric Theory of Functions [in Russian], Far-East University, Vladivostok (2003).

    Google Scholar 

  15. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo (1959).

    MATH  Google Scholar 

  16. A. K. Bakhtin, “Inequalities for inner radii of nonoverlapping domains and open sets,” Dopov. Nats. Akad. Nauk Ukr., No. 10, 7–13 (2006).

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, Ukraine

    A. K. Bakhtin

Authors
  1. A. K. Bakhtin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1601–1618, December, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bakhtin, A.K. Sharp estimates for inner radii of systems of nonoverlapping domains and open sets. Ukr Math J 59, 1800–1818 (2007). https://doi.org/10.1007/s11253-008-0027-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-008-0027-6

Keywords

Search

Navigation