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On composition principles for reduced moduli

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Abstract

We obtain the necessary and sufficient conditions for equality holding in the composition principles for generalized reduced moduli. As an application, we give some descriptions of extremal configurations in known and new inequalities for products of the Robin radii of nonoverlapping domains.

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References

  1. Kuz’mina G. V., “Methods of the geometric theory of functions. I, II,” St. Petersburg Math. J., 9, No. 3 and 5, 455–507 and 889–930 (1998).

    MathSciNet  Google Scholar 

  2. Solynin A. Yu., “Moduli and extremal metric problems,” St. Petersburg Math. J., 11, No. 1, 1–65 (2000).

    MathSciNet  Google Scholar 

  3. Emel’yanov E. G., “On quadratic differentials in multiply connected domains that are perfect squares. II,” Zap. Nauchn. Sem. POMI, 350, 40–51 (2007).

    Google Scholar 

  4. Dubinin V. N., Capacities of Condensers and Symmetrization in Geometric Theory of Functions of a Complex Variable [in Russian], Dal’nauka, Vladivostok (2009).

  5. Dubinin V. N., “Generalized condensers and the asymptotics of their capacities under degeneration of some plates,” J. Math. Sci. (New York), 129, No. 3, 3835–3842 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  6. Dubinin V. N. and Kirillova D. A., “On extremal decomposition problems,” J. Math. Sci. (New York), 157, No. 4, 573–583 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  7. Dubinin V. N. and Eyrikh N. V., “Reduced generalized module,” Dal’nevostochn. Mat. Zh., 3, No. 2, 135–147 (2002).

    Google Scholar 

  8. Dubinin V. N. and Prilepkina E. G., “Preservation of the generalized reduced module under some geometric transformations of the plane domains,” Dalnevostochn. Mat. Zh., 6, No. 1–2, 39–56 (2005).

    Google Scholar 

  9. Karp D. and Prilepkina E., “Reduced modules with free boundary and its applications,” Ann. Acad. Sci. Fenn., Math., 34, 353–378 (2009).

    MATH  MathSciNet  Google Scholar 

  10. Dubinin V. N. and Eyrikh N. V., “Some applications of generalized condensers to analytic function theory,” J. Math. Sci. (New York), 133, No. 6, 1634–1647 (2006).

    Article  Google Scholar 

  11. Dubinin V. N. and Kovalev L. V., “The reduced modulus of the complex sphere,” J. Math. Sci. (New York), 105, No. 4, 2165–2179 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  12. Prilepkina E. G., “Distortion theorems for univalent functions in multiply connected domains,” Dalnevostochn. Mat. Zh., 9, No. 1–2, 140–149 (2009).

    MathSciNet  Google Scholar 

  13. Duren P., Pfaltzgraff J., and Thurman E., “Physical interpretation and further properties of Robin capacity,” St. Petersburg Math. J., 9, No. 3, 607–614 (1998).

    MathSciNet  Google Scholar 

  14. Duren P. L. and Schiffer M. M., “Robin functions and energy functionals of multiply connected domains,” Pacific J. Math., 148, No. 2, 251–273 (1991).

    MATH  MathSciNet  Google Scholar 

  15. O’Neill M. D. and Thurman R. E., “Extremal domains for Robin capacity,” Complex Variables, 41, 91–109 (2000).

    MATH  MathSciNet  Google Scholar 

  16. Nasyrov S., “Robin capacity and lift of infinitely thin airfoils,” Complex Variables, 47, No. 2, 93–107 (2002).

    MATH  MathSciNet  Google Scholar 

  17. Stiemer M., “A representation formula for the Robin function,” Complex Variables, 48,No. 5, 417–427 (2003).

    MATH  MathSciNet  Google Scholar 

  18. Vasil’ev A. Yu., “Robin’s modulus in a Hele-Shaw problem,” Complex Variables, 49, No. 7–9, 663–672 (2004).

    MATH  Google Scholar 

  19. Gaier D. and Hayman W., “On the computation of modules of long quadrilaterals,” Constr. Approx., 7, 453–467 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  20. Hayman W. K., Multivalent Functions [Russian translation], Izdat. Inostr. Lit., Moscow (1960).

    Google Scholar 

  21. Kuz’mina G. V., “On extremal properties of quadratic differentials with strip domains in the structure of trajectories,” Zap. Nauchn. Sem. POMI, 154, 110–129 (1986).

    MATH  Google Scholar 

  22. Emel’yanov E. G., “Problems of extremal decomposition,” J. Soviet Math., 43, No. 4, 2558–2566 (1988).

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  24. Nasyrov S. R., “Variations of Robin capacity and applications,” Siberian Math. J., 49, No. 5, 894–910 (2008).

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Institute of Applied Mathematics, Vladivostok, Russia

    E. G. Prilepkina

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  1. E. G. Prilepkina
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Correspondence to E. G. Prilepkina.

Additional information

Original Russian Text Copyright © 2011 Prilepkina E. G.

The author was supported by the Russian Foundation for Basic Research (Grant 08-01-00028) and the Far East Division of the Russian Academy of Sciences (Grant 09-III-A-01-008).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 6, pp. 1357–1372, November–December, 2011.

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Prilepkina, E.G. On composition principles for reduced moduli. Sib Math J 52, 1079–1091 (2011). https://doi.org/10.1134/S0037446611060139

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  • DOI: https://doi.org/10.1134/S0037446611060139

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