Advertisement

Siberian Mathematical Journal

, Volume 52, Issue 6, pp 1079–1091 | Cite as

On composition principles for reduced moduli

  • E. G. Prilepkina
Article

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.

Keywords

reduced modulus Robin function Neumann function nonoverlapping domain extremal partition problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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).MathSciNetGoogle Scholar
  2. 2.
    Solynin A. Yu., “Moduli and extremal metric problems,” St. Petersburg Math. J., 11, No. 1, 1–65 (2000).MathSciNetGoogle Scholar
  3. 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. 4.
    Dubinin V. N., Capacities of Condensers and Symmetrization in Geometric Theory of Functions of a Complex Variable [in Russian], Dal’nauka, Vladivostok (2009).Google Scholar
  5. 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).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Dubinin V. N. and Kirillova D. A., “On extremal decomposition problems,” J. Math. Sci. (New York), 157, No. 4, 573–583 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dubinin V. N. and Eyrikh N. V., “Reduced generalized module,” Dal’nevostochn. Mat. Zh., 3, No. 2, 135–147 (2002).Google Scholar
  8. 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. 9.
    Karp D. and Prilepkina E., “Reduced modules with free boundary and its applications,” Ann. Acad. Sci. Fenn., Math., 34, 353–378 (2009).zbMATHMathSciNetGoogle Scholar
  10. 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).CrossRefGoogle Scholar
  11. 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).CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Prilepkina E. G., “Distortion theorems for univalent functions in multiply connected domains,” Dalnevostochn. Mat. Zh., 9, No. 1–2, 140–149 (2009).MathSciNetGoogle Scholar
  13. 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).MathSciNetGoogle Scholar
  14. 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).zbMATHMathSciNetGoogle Scholar
  15. 15.
    O’Neill M. D. and Thurman R. E., “Extremal domains for Robin capacity,” Complex Variables, 41, 91–109 (2000).zbMATHMathSciNetGoogle Scholar
  16. 16.
    Nasyrov S., “Robin capacity and lift of infinitely thin airfoils,” Complex Variables, 47, No. 2, 93–107 (2002).zbMATHMathSciNetGoogle Scholar
  17. 17.
    Stiemer M., “A representation formula for the Robin function,” Complex Variables, 48,No. 5, 417–427 (2003).zbMATHMathSciNetGoogle Scholar
  18. 18.
    Vasil’ev A. Yu., “Robin’s modulus in a Hele-Shaw problem,” Complex Variables, 49, No. 7–9, 663–672 (2004).zbMATHGoogle Scholar
  19. 19.
    Gaier D. and Hayman W., “On the computation of modules of long quadrilaterals,” Constr. Approx., 7, 453–467 (1991).CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hayman W. K., Multivalent Functions [Russian translation], Izdat. Inostr. Lit., Moscow (1960).Google Scholar
  21. 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).zbMATHGoogle Scholar
  22. 22.
    Emel’yanov E. G., “Problems of extremal decomposition,” J. Soviet Math., 43, No. 4, 2558–2566 (1988).CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Goluzin G. M., Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1966).Google Scholar
  24. 24.
    Nasyrov S. R., “Variations of Robin capacity and applications,” Siberian Math. J., 49, No. 5, 894–910 (2008).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Institute of Applied MathematicsVladivostokRussia

Personalised recommendations