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Application of a separating transformation to estimates of inner radii of open sets

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Abstract

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets.

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References

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

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

    Google Scholar 

  3. J. A. Jenkins, Univalent Functions and Conformal Mappings, Springer, Berlin (1958).

    Google Scholar 

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

    MATH  Google Scholar 

  5. 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 

  6. G. P. Bakhtina, Variational Methods and Quadratic Differentials in Problems of Nonoverlapping Domains [in Russian], Author’s Abstract of the Doctoral Degree Thesis (Physics and Mathematics), Kiev (1975).

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

    Google Scholar 

  8. 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 

  9. V. N. Dubinin, “Asymptotics of the modulus of a degenerate capacitor and some applications,” Zap. Nauchn. Sem. LOMI, 237, 56–73 (1997).

    Google Scholar 

  10. V. N. Dubinin, Capacitances of Capacitors in Geometric Theory of Functions [in Russian], Far-East University, Vladivostok (2003).

    Google Scholar 

  11. V. N. Dubinin, “On the product of inner radii of ‘partially nonoverlapping’ domains,” in: Problems of the Metric Theory of Mappings and Its Applications [in Russian], Naukova Dumka, Kiev (1978), pp. 24–31.

    Google Scholar 

  12. E. G. Emel’yanov, “On relationship between two problems of extremal decomposition,” Zap. Nauchn. Sem. LOMI, 160, 91–98 (1987).

    MATH  MathSciNet  Google Scholar 

  13. A. K. Bakhtin, “Extremal problems of nonoverlapping domains with free poles on two circles,” Dopov. Nats. Akad. Nauk Ukr., No. 7, 12–16 (2005).

  14. A. K. Bakhtin, “Reduced moduli of open sets and extremal problems with free poles,” Dopov. Nats. Akad. Nauk Ukr., No. 5, 7–13 (2006).

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

  16. A. K. Bakhtin and V. E. V’yun, “Separating transformation and inequalities for open sets,” Dopov. Nats. Akad. Nauk Ukr., No. 4, 7–11 (2007).

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Authors and Affiliations

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

    A. K. Bakhtin & V. E. V’yun

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  1. A. K. Bakhtin
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  2. V. E. V’yun
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1313–1321, October, 2007.

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Bakhtin, A.K., V’yun, V.E. Application of a separating transformation to estimates of inner radii of open sets. Ukr Math J 59, 1472–1481 (2007). https://doi.org/10.1007/s11253-008-0003-1

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  • DOI: https://doi.org/10.1007/s11253-008-0003-1

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