Skip to main content
Log in

Non-real zeros of derivatives of meromorphic functions

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. 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.

    Article  MathSciNet  MATH  Google Scholar 

  2. 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.

    Article  MathSciNet  MATH  Google Scholar 

  3. 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Anal. Math., 91 (2003), 353–367.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Craven, G. Csordas, and W. Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2), 125 (1987), 405–431.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Edrei and W. H. J. Fuchs, Bounds for the number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc. (3), 12 (1962), 315–344.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Edrei and W. H. J. Fuchs, On meromorphic functions with regions free of poles and zeros, Acta Math., 108 (1962), 113–145.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Frank, Über die Nullstellen von linearen Differentialpolynomen mit meromorphen Koeffizienten, Complex Methods on Partial Differential Equations, Akademie-Verlag, Berlin, 1989, pp. 39–48.

    Google Scholar 

  9. G. Frank, W. Hennekemper, and G. Polloczek, Über die Nullstellen meromorpher Funktionen und deren Ableitungen, Math. Ann. 225 (1977), 145–154.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Frank and G. Weissenborn, Rational deficient functions of meromorphic functions, Bull. London Math. Soc., 18 (1986), 29–33.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Frank and G. Weissenborn, On the zeros of linear differential polynomials of meromorphic functions, Complex Variables Theory Appl., 12 (1989), 77–81.

    MathSciNet  MATH  Google Scholar 

  12. A. A. Gol’dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Nauka, Moscow, 1970 (Russian); English transl., Amer. Math. Soc. Providence, RI, 2008.

    Google Scholar 

  13. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2), 37 (1988), 88–104.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

    MATH  Google Scholar 

  15. 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.

    Article  MathSciNet  MATH  Google Scholar 

  16. W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull., 17 (1974), 317–358.

    Article  MathSciNet  MATH  Google Scholar 

  17. W. K. Hayman, Subharmonic Functions Vol. 2, Academic Press, London, 1989.

    MATH  Google Scholar 

  18. S. Hellerstein, L.-C. Shen, and J. Williamson, Reality of the zeros of an entire function and its derivatives, Trans. Amer.Math. Soc., 275 (1983), 319–331.

    Article  MathSciNet  MATH  Google Scholar 

  19. 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.

    Article  MathSciNet  MATH  Google Scholar 

  20. 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.

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, MA, 1969.

    MATH  Google Scholar 

  22. E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.

    MATH  Google Scholar 

  23. A. Hinkkanen, Iteration and the zeros of the second derivative of a meromorphic function, Proc. London Math. Soc. (3), 65 (1992), 629–650.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Hinkkanen, Reality of zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math., 22 (1997), 21–38.

    MathSciNet  MATH  Google Scholar 

  25. A. Hinkkanen, Zeros of derivatives of strictly non-real meromorphic functions, Ann. Acad. Sci. Fenn. Math., 22 (1997), 39–74.

    MathSciNet  MATH  Google Scholar 

  26. A. Hinkkanen, Iteration, level sets, and zeros of derivatives of meromorphic functions, Ann. Acad. Sci. Fenn. Math., 23 (1998), 317–388.

    MathSciNet  MATH  Google Scholar 

  27. A. Hinkkanen and J. F. Rossi, On a problem of Hellerstein, Shen and Williamson, Proc. Amer. Math. Soc., 92 (1984), 72–74.

    Article  MathSciNet  MATH  Google Scholar 

  28. 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.

    Article  MathSciNet  MATH  Google Scholar 

  29. J. K. Langley, Proof of a conjecture of Hayman concerning f and f″, J. London Math. Soc. (2), 48 (1993), 500–514.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. K. Langley, On second order linear differential polynomials, Results Math., 26 (1994), 51–82.

    Article  MathSciNet  MATH  Google Scholar 

  31. J. K. Langley, The second derivative of a meromorphic function of finite order, Bull. London Math. Soc., 35 (2003), 97–108.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. K. Langley, Non-real zeros of higher derivatives of real entire functions of infinite order, J. Anal. Math., 97 (2005), 357–396.

    Article  MathSciNet  MATH  Google Scholar 

  33. J. K. Langley, Non-real zeros of derivatives of real meromorphic functions, Proc. Amer. Math. Soc., 137 (2009), 3355–3367.

    Article  MathSciNet  MATH  Google Scholar 

  34. J. K. Langley, Zeros of derivatives of meromorphic functions, Comput. Methods Funct. Theory, 10 (2010), 421–439.

    Article  MathSciNet  MATH  Google Scholar 

  35. J. K. Langley, Non-real zeros of real differential polynomials, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 631–639.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. K. Langley, Zeros of derivatives of real meromorphic functions, Comput. Methods Funct. Theory, 12 (2012), 241–256.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. K. Langley, The reciprocal of a real entire function and non-real zeros of higher derivatives, Ann. Acad. Sci. Fenn. Math., 38 (2013), 855–871.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. K. Langley, Derivatives of meromorphic functions of finite order, Comput. Methods Funct. Theory, 14 (2014), 195–207.

    Article  MathSciNet  MATH  Google Scholar 

  39. B. Ja. Levin, Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956; 2-nd English transl., Amer. Math. Soc., Providence, RI, 1980.

    MATH  Google Scholar 

  40. 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., AMS Transl. (2) 32 (1963), 323–357.

    MathSciNet  Google Scholar 

  41. J. Lewis, J. Rossi, and A. Weitsman, On the growth of subharmonic functions along paths, Ark. Mat., 22 (1984), 109–119.

    Article  MathSciNet  MATH  Google Scholar 

  42. P. Lounesto and S. Toppila, On a conjecture of Hellerstein, Shen and Williamson, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 40 (1986), 267–268.

    MathSciNet  MATH  Google Scholar 

  43. R. Nevanlinna, Eindeutige analytische Funktionen, 2te Aufl., Springer, Berlin, 1953.

    Book  MATH  Google Scholar 

  44. D. A. Nicks, Real meromorphic functions and a result of Hinkkanen and Rossi, Illinois J. Math., 53 (2009), 605–622.

    MathSciNet  MATH  Google Scholar 

  45. C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.

    Book  MATH  Google Scholar 

  46. 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.

    MathSciNet  MATH  Google Scholar 

  47. W. Schwick, Normality criteria for families of meromorphic functions, J. Anal. Math., 52 (1989), 241–289.

    Article  MathSciNet  MATH  Google Scholar 

  48. 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.

    Article  MathSciNet  MATH  Google Scholar 

  49. L. C. Shen, Construction of a differential equation y″ + Ay = 0 with solutions having prescribed zeros, Proc. Amer. Math. Soc., 95 (1985), 544–546.

    MathSciNet  MATH  Google Scholar 

  50. N. Steinmetz, On the zeros of a certain Wronskian, Bull. London Math. Soc., 20 (1988), 525–531.

    Article  MathSciNet  MATH  Google Scholar 

  51. N. Steinmetz, Rational Iteration, Walter de Gruyter, Berlin, 1993.

    Book  MATH  Google Scholar 

  52. M. Tsuji, On Borel’s directions of meromorphic functions of finite order, I, Tôhoku Math. J., 2 (1950), 97–112.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

    MATH  Google Scholar 

  54. G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949.

    Google Scholar 

  55. L. Zalcman, Normal families: new perspectives, Bull. Amer. Math. Soc. (N.S), 35 (1998), 215–230.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. K. Langley.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Langley, J.K. Non-real zeros of derivatives of meromorphic functions. JAMA 133, 183–228 (2017). https://doi.org/10.1007/s11854-017-0031-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-017-0031-6

Navigation