Journal d'Analyse Mathématique

, Volume 119, Issue 1, pp 365–402 | Cite as

Gol’dberg’s constants

Article

Abstract

We study two extremal problems of geometric function theory introduced by A. A. Gol’dberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process, we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them, and make a conjecture. Gol’dberg’s problems have important applications to control theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, New York, 1978.Google Scholar
  2. [2]
    L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory, McGraw-Hill, New York, 1973.MATHGoogle Scholar
  3. [3]
    N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Amer. Math. Soc., Providence, RI, 1990.MATHGoogle Scholar
  4. [4]
    C. M. Baribaud, Closed geodesics on pairs of pants, Israel J. Math. 109 (1999), 339–347.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    P. Batra, On small circles containing zeros and ones of analytic functions, Complex Variables Theory Appl. 49 (2004), 787–791.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    P. Batra, On Gol’dberg’s constant A 2, Comput. Methods Funct. Theory 7 (2007), 33–41.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    V. Blondel, Simultaneous Stabilization of Linear Systems, Springer, Berlin, 1994.MATHCrossRefGoogle Scholar
  8. [8]
    V. D. Blondel, R. Rupp, and H. S. Shapiro, On zero and one points of analytic functions, Complex Variables Theory Appl. 28 (1995), 189–192.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    P. R. Brown and R. M. Porter, Conformal mapping of circular quadrilaterals and Weierstrass elliptic functions, Comput. Methods Funct. Theory 11 (2011), 463–486.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J. Burke, D. Henrion, A. Lewis, and M. Overton, Stabilization via nonsmooth, nonconvex optimization, IEEE Trans. Automat. Control 51 (2006), 1760–1769.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Y. J. Chang and N. V. Sahinidis, Global optimization in stabilizing controller design, J. Global Optim. 38 (2007), 509–526.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    E. Chirka, On the propagation of holomorphic motions, Dokl. Akad. Nauk 397 (2004), 37–40.MathSciNetGoogle Scholar
  13. [13]
    V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Russian Math. Surveys 49 (1994), 1–79.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    A. Eremenko, On the hyperbolic metric of the complement of a rectangular lattice, preprint, arXiv:1110.2696.Google Scholar
  15. [15]
    B. Fine, Trace classes and quadratic forms in the modular group, Canad. Math. Bull. 37 (1994), 202–212.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A. A. Gol’dberg, On a theorem of Landau’s type, Teor. Funkciĭ Funkcional. Anal. i Priložen. 17 (1973), 200–206.MATHGoogle Scholar
  17. [17]
    J. A. Hempel and S. J. Smith, Hyperbolic lengths of geodesics surrounding two punctures, Proc. Amer. Math. Soc. 103 (1988), 513–516.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    A. Hurwitz, Über die Anwendung der elliptischen Modulfunktionen auf einem Satz der allgemeinen Funktionentheorie, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 49 (1904), 242–253.Google Scholar
  19. [19]
    A. Hurwitz and R. Courant, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Springer, Berlin, 1964.MATHCrossRefGoogle Scholar
  20. [20]
    E. L. Ince, Ordinary Differential Equations, Longmans, Green and Co., London, 1926; Dover reprint, 1956.Google Scholar
  21. [21]
    J. Jenkins, On a problem of A. A. Gol’dberg, Ann. Univ. Mariae Curie-Skłodowska Sect. A 36/37 (1982/83), 83–86.MathSciNetGoogle Scholar
  22. [22]
    M. Lawrentjew and B. Schabat, Methoden der komplexen Funktionentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1967.MATHGoogle Scholar
  23. [23]
    O. Lehto and K. Virtanen, Quasikonforme Abbildungen, Springer, Berlin, 1965. English translation: Springer, 1973.MATHGoogle Scholar
  24. [24]
    Z. Nehari, The elliptic modular function and a class of analytic functions first considered by Hurwitz, Amer. J. Math. 69 (1947), 70–86.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    V. V. Patel, G. Deodhare, and T. Viswanath, Some applications of randomized algorithms for control system design, Automatica, 38 (2002), 2085–2092.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    P. Schmutz Schaller, The modular torus has maximal length spectrum, Geom. Funct. Anal. 6 (1996), 1057–1073.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347–355.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Bin Wang and Xinyun Zhu, On the traces of elements of modular group, Linear Algebra Appl. 438 (2013), 604–608.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2013

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrecht-Universität zu KielKielGermany
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations