Journal d'Analyse Mathématique

, Volume 117, Issue 1, pp 87–118 | Cite as

Non-real zeroes of real entire derivatives

  • Daniel A. Nicks


A real entire function belongs to the Laguerre-Pólya class LP if it is the limit of a sequence of real polynomials with real zeroes. By building upon results that resolved a long-standing conjecture of Wiman, a number of conditions are established under which a real entire function f must belong to the class LP, or to one of the related classes U 2p * . These conditions typically involve the non-real zeroes of f and its derivatives, or those of the differential polynomial f f″−a(f′)2.


Entire Function Meromorphic Function Logarithmic Derivative Real Zero Distinct Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    M. Ålander, Sur les zéros extraordinaires des dérivées des fonctions entières réelles, Ark. för Mat., Astron. och Fys. 11 (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 (1922), 1–19.Google Scholar
  3. [3]
    W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 151–188.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    W. Bergweiler, On the zeros of certain homogeneous differential polynomials, Arch. Math. 64 (1995), 199–202.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    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
  6. [6]
    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
  7. [7]
    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
  8. [8]
    W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Anal. Math. 91 (2003), 353–367.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    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
  10. [10]
    A. Fletcher, J. K. Langley and J. Meyer, Nonvanishing derivatives and the MacLane class A, Illinois J. Math. 53 (2009), 379–390.MathSciNetMATHGoogle Scholar
  11. [11]
    A. A. Gol’dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Amer. Math. Soc., Providence RI, 2008.Google Scholar
  12. [12]
    G. 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
  13. [13]
    W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  14. [14]
    W. K. Hayman, On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. (3) 14A (1965), 93–128.MathSciNetCrossRefGoogle Scholar
  15. [15]
    S. Hellerstein and J. Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249.MathSciNetMATHGoogle Scholar
  16. [16]
    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
  17. [17]
    S. Hellerstein and C. C. Yang, Half-plane Tumura-Clunie theorems and the real zeros of successive derivatives, J. London Math. Soc. (2) 4 (1972), 469–481.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    J. D. Hinchliffe, The Bergweiler-Eremenko theorem for finite lower order, Result. Math. 43 (2003), 121–128.MathSciNetMATHGoogle Scholar
  19. [19]
    A. Hinkkanen, Iteration and the zeros of the second derivative of a meromorphic function, Proc. London Math. Soc. (3) 65 (1992), 629–650.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    E. Laguerre, Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci. Paris 95 (1882); Oeuvres 1 174–177.Google Scholar
  21. [21]
    J. K. Langley, A lower bound for the number of zeros of a meromorphic function and its second derivative, Proc. Edinburgh Math. Soc. 39 (1996), 171–185.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    J. K. Langley, Non-real zeros of higher derivatives of real entire functions of infinite order, J. Anal. Math. 97 (2005), 357–396.MathSciNetCrossRefGoogle Scholar
  23. [23]
    J. K. Langley, Solution of a problem of Edwards and Hellerstein, Comput. Methods Funct. Theory 6 (2006), 243–252.MathSciNetMATHGoogle Scholar
  24. [24]
    J. K. Langley, Non-real zeros of linear differential polynomials, J. Anal. Math. 107 (2009), 107–140.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    J. K. Langley, Non-real zeros of real differential polynomials, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 631–639.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    J. K. Langley, Non-real zeros of derivatives of real meromorphic functions of infinite order, Math. Proc. Cambridge Philos. Soc. 150 (2011), 343–351.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    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., Amer. Math. Soc. Transl. (2) 32 (1963), 323–357.MathSciNetMATHGoogle Scholar
  28. [28]
    B. Ja. Levin, Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956. 2nd English transl., Amer. Math. Soc., Providence RI, 1980.MATHGoogle Scholar
  29. [29]
    J. Milnor, Dynamics in One Complex Variable, 3rd edition, Princeton University Press, Princeton, 2006.MATHGoogle Scholar
  30. [30]
    R. Nevanlinna, Eindeutige analytische Funktionen, 2nd edition, Springer-Verlag, Berlin, 1953.MATHGoogle Scholar
  31. [31]
    G. Pólya, Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279–295.MATHCrossRefGoogle Scholar
  32. [32]
    G. Pólya, Sur une question concernant les fonctions entières, C. R. Acad. Sci. Paris 158 (1914), 330–333.MATHGoogle Scholar
  33. [33]
    T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. (2) 129 (1989), 179–193.MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    L. C. Shen, Influence of the distribution of the zeros of an entire function and its second derivative on the growth of the function, J. London Math. Soc. (2) 31 (1985), 305–320.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    M. Tsuji, On Borel’s directions of meromorphic functions of finite order, I, Tôhoku Math. J. 2 (1950), 97–112.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    M. Tsuji, Potential Theory in Modern Function Theory, 2nd edition, Chelsea Publishing Co., New York, 1975.MATHGoogle Scholar

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© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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