Computational Methods and Function Theory

, Volume 10, Issue 2, pp 421–439 | Cite as

Zeros of Derivatives of Meromorphic Functions

  • James Langley


The first part of this paper is an expanded version of a plenary lecture of the same title, given by the author at the CMFT conference at Bilkent University, Ankara, in June 2009. In the second part of the paper, a considerably stronger version of one of the main results is proved.


Meromorphic functions zeros of derivatives non-real zeros 

2000 MSC



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Ålander, Sur les zéros extraordinaires des dérivées des fonctions entières réelles, Ark. för Mat., Astron. och Fys 11 no.15 (1916), 1–18.Google Scholar
  2. 2.
    M. Ålander, Sur les zéros complexes des dérivées des fonctions entières réelles, Ark. för Mat., Astron. och Fys 16 no.10 (1922), 1–19Google Scholar
  3. 3.
    W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11 (1995), 355–373.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    W. Bergweiler and A. Eremenko, Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions, Acta Math. 197 (2006), 145–166.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    W. Bergweiler, A. Eremenko and J. K. Langley, Real entire functions of infinite order and a conjecture of Wiman, Geom. Funct. Anal. 13 (2003), 975–991.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Analyse Math. 91 (2003), 353–367.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    F. Brüggemann, Proof of a conjecture of Frank and Langley concerning zeros of meromorphic functions and linear differential polynomials, Analysis 12 no.1/2 (1992), 5–30.MathSciNetMATHGoogle Scholar
  8. 8.
    T. Craven, G. Csordas and W. Smith, Zeros of derivatives of entire functions, Proc. Amer. Math. Soc. 101 (1987), 323–326.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    T. Craven, G. Csordas and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Annals of Math. (2) 125 (1987), 405–431.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    S. Edwards and S. Hellerstein, Non-real zeros of derivatives of real entire functions and the Pólya-Wiman conjectures, Complex Var. Theory Appl. 47 (2002), 25–57.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Zeit. 149 (1976), 29–36.MATHCrossRefGoogle Scholar
  12. 12.
    G. Frank, Über die Nullstellen von linearen Differentialpolynomen mit meromorphen Ko-effizienten, Complex Methods on Partial Differential Equations, 39-48, Math. Res. 53, Akademie-Verlag, Berlin 1989.Google Scholar
  13. 13.
    G. Frank and S. Hellerstein, On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients, Proc. London Math. Soc. (3) 53 (1986), 407–428.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    G. Frank, W. Hennekemper and G. Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), 145–154.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis 19 (1999), 173–194.MathSciNetMATHGoogle Scholar
  16. 16.
    G. Frank and G. Weissenborn, Rational deficient functions of meromorphic functions, Bull. London Math. Soc. 18 (1986), 29–33.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    A. A. Gol’dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions, Nauka, Moscow, 1970 (Russian); English translation, Translations of Mathematical Monographs 236, Amer. Math. Soc. Providence 2008.Google Scholar
  18. 18.
    G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88–104.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    W. K. Hayman, Picard values of meromorphic functions and their derivatives, Annals of Math. 70 (1959), 9–42.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  21. 21.
    W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317–358.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249.MathSciNetMATHGoogle Scholar
  23. 23.
    S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, II, Trans. Amer. Math. Soc. 234 (1977), 497–503.MathSciNetMATHGoogle Scholar
  24. 24.
    S. Hellerstein and J. Williamson, The zeros of the second derivative of the reciprocal of an entire function, Trans. Amer. Math. Soc. 263 (1981), 501–513.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    S. Hellerstein, L.-C. Shen and J. Williamson, Reality of the zeros of an entire function and its derivatives, Trans. Amer. Math. Soc. 273 (1983), 319–331.MathSciNetCrossRefGoogle Scholar
  26. 26.
    S. Hellerstein, L.-C. Shen and J. Williamson, Real zeros of derivatives of meromorphic functions and solutions of second order differential equations, Trans. Amer. Math. Soc. 285 (1984), 759–776.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. 22 (1997), 1–38.Google Scholar
  28. 28.
    A. Hinkkanen, Zeros of derivatives of strictly non-real meromorphic functions, Ann. Acad. Sci. Fenn. 22 (1997), 39–74.MathSciNetMATHGoogle Scholar
  29. 29.
    A. Hinkkanen, Iteration, level sets, and zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. 23 (1998), 317–388.MathSciNetMATHGoogle Scholar
  30. 30.
    K. Ishizaki, Some remarks on results of Mues about deficiency sums of derivatives, Arch. Math. (Basel) 55 (1990), 374–379.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    H. Ki and Y.-O. Kim, On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture, Duke Math. J. 104 (2000), 45–73.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Y.-O. Kim, A proof of the Pólya-Wiman conjecture, Proc. Amer. Math. Soc. 109 (1990), 1045–1052.MathSciNetMATHGoogle Scholar
  33. 33.
    E. Laguerre, Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci. Paris 95 (1882); Oevres, t. 1 174–177.Google Scholar
  34. 34.
    I. Laine, Nevanlinna theory and complex differential equations, de Gruyter Studies in Math. 15, Walter de Gruyter, Berlin/New York, 1993.CrossRefGoogle Scholar
  35. 35.
    J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. London Math. Soc. (2) 48 (1993), 500–514.MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    J. K. Langley, On second order linear differential polynomials, Result. Math. 26 (1994), 51–82.MathSciNetMATHGoogle Scholar
  37. 37.
    J. K. Langley, The second derivative of a meromorphic function of finite order, Bulletin London Math. Soc. 35 (2003), 97–108.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    J. K. Langley, Non-real zeros of higher derivatives of real entire functions of infinite order, J. Analyse Math. 97 (2005), 357–396.MathSciNetCrossRefGoogle Scholar
  39. 39.
    J. K. Langley, Non-real zeros of linear differential polynomials, J. Analyse Math. 107 (2009), 107–140.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    J. K. Langley, Logarithmic singularities and the zeros of the second derivative, Comput. Methods Funct. Theory 9 no.2 (2009, 565–578.MathSciNetMATHGoogle Scholar
  41. 41.
    —, Non-real zeros of derivatives of real meromorphic functions, to appear in Proc. Amer. Math. Soc. Google Scholar
  42. 42.
    B. Ja. Levin, Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956. 2-nd English transl., AMS, Providence RI, 1980.Google Scholar
  43. 43.
    B. Ja. Levin and I. V. Ostrovskii, The dependence of the growth of an entire function on the distribution of zeros of its derivatives, Sibirsk. Mat. Zh. 1 (1960), 427–455; English transl. in AMS Transl. (2) 32 (1963), 323–357.MathSciNetMATHGoogle Scholar
  44. 44.
    E. Mues, Über eine Defekt- und Verzweigungsrelation für die Ableitung meromorpher Funktionen, Manuscripta Math. 5 (1971), 275–297.MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    R. Nevanlinna, Eindeutige analytische Funktionen, 2. Aufl., Springer, Berlin, 1953.MATHCrossRefGoogle Scholar
  46. 46.
    D. A. Nicks, Ph.D. thesis, University of Nottingham, to appear.Google Scholar
  47. 47.
    —, Real meromorphic functions and a result of Hinkkanen and Rossi, to appear in Illinois J. Math. Google Scholar
  48. 48.
    G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279–295.MATHCrossRefGoogle Scholar
  49. 49.
    G. Pólya, Sur une question concernant les fonctions entières, C. R. Acad. Sci. Paris 158 (1914), 330–333.MATHGoogle Scholar
  50. 50.
    G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Zeit. 12 (1922), 36–60.MATHCrossRefGoogle Scholar
  51. 51.
    G. Pólya, On the zeros of the derivatives of a function and its analytic character, Bulletin Amer. Math. Soc. 49 (1943), 178–191.CrossRefGoogle Scholar
  52. 52.
    J. Rossi, The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative, Trans. Amer. Math. Soc. 270 (1982), 667–683.MathSciNetMATHGoogle Scholar
  53. 53.
    W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241–289.MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Annals of Math. 129 (1989), 179–193.MathSciNetMATHCrossRefGoogle Scholar
  55. 55.
    Steinmetz, N 1987On the zeros of \((f^{(p)}+a_{p-1}f^{(p-1)}+\dots +a_{0}f)f\) Analysis7375389MathSciNetMATHGoogle Scholar
  56. 56.
    N. Steinmetz, On the zeros of a certain Wronskian, Bull. London Math. Soc. 20 (1988), 525–531.MathSciNetMATHCrossRefGoogle Scholar
  57. 57.
    M. Tsuji, On Borel’s directions of meromorphic functions of finite order, I, Tôhoku Math. J. 2 (1950), 97–112.MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Y. F. Wang, On Mues’ conjecture and Picard values, Sci. China Ser. A 36 (1993), 28–35.MathSciNetMATHGoogle Scholar
  59. 59.
    Y. F. Wang and L. Yang, Drasin’s problems and Mues’ conjecture, Sci. China Ser. A 35 no.10 (1992), 1180–1190.MathSciNetMATHGoogle Scholar
  60. 60.
    L. Yang, Precise estimate of total deficiency of meromorphic derivatives, J. Analyse Math. 55 (1990), 287–296.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Heldermann  Verlag 2010

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamUK

Personalised recommendations