Skip to main content

On the behaviour of the successive derivatives of meromorphic functions on the final set

Journal d'Analyse Mathématique volume 120, pages 131–149 (2013)Cite this article

Abstract

We study the behaviour of the sequence of successive derivatives of meromorphic functions at points of the so-called final set. We prove that, whereas in many cases this sequence tends to ∞, for a special class of meromorphic functions, it may have extremely wild behaviour. We also prove a connection between the derivatives of meromorphic functions from this class and so-called Dirichlet sets.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. P. Beise, T. Meyrath, and J. Müller, Universality properties of Taylor series inside the domain of holomorphy, J. Math. Anal. Appl. 383 (2011), 234–238.

    MathSciNet  Article  MATH  Google Scholar 

  2. A. Edrei and G. R. MacLane, On the zeros of the derivatives of an entire function, Proc. Amer. Math. Soc. 8 (1957), 702–706.

    MathSciNet  MATH  Google Scholar 

  3. R. Gethner, On the zeros of the derivatives of some entire functions of finite order, Proc. Edinburgh Math. Soc. 28 (1985), 381–407.

    MathSciNet  Article  Google Scholar 

  4. R. Gethner, Accumulation rate of zeros of the successive derivatives of meromorphic functions, J. London Math. Soc. (2) 33 (1986), 294–302.

    MathSciNet  Article  MATH  Google Scholar 

  5. R. Gethner, Zeros of the successive derivatives of Hadamard gap series, Trans. Amer. Math. Soc. 339 (1993), 799–807.

    MathSciNet  Article  MATH  Google Scholar 

  6. R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281–288.

    MathSciNet  Article  MATH  Google Scholar 

  7. K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. 36 (1999), 345–381.

    MathSciNet  Article  MATH  Google Scholar 

  8. K. G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Springer, London, 2011.

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  10. W. K. Hayman and A. Hinkkanen, Normal families and the final set, Complex Var. Theory Appl. 43 (2001), 495–504.

    MathSciNet  Article  MATH  Google Scholar 

  11. A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1995.

    Book  MATH  Google Scholar 

  12. T. W. Körner, Kahane's Helson curve, J. Fourier Anal. Appl. Special Issue Orsay 1993 (1995), 325–346.

    Google Scholar 

  13. G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math. 2 (1952), 72–87.

    MathSciNet  Article  MATH  Google Scholar 

  14. G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Z. 12 (1922), 36–60.

    MathSciNet  Article  MATH  Google Scholar 

  15. G. Pólya, On the zeros of the derivatives of a function and its analytic character, Bull. Amer. Math. Soc. 49 (1943), 178–191.

    MathSciNet  Article  Google Scholar 

  16. G. Pólya, On the zeros of successive derivatives — an example, J. Analyse Math. 30 (1976), 452–455.

    MathSciNet  Article  MATH  Google Scholar 

  17. C. L. Prather and J. K. Shaw, A Pólya 'shire’ theorem for functions with algebraic singularities, Internat. J. Math. Math. Sci. 5 (1982), 691–706.

    MathSciNet  Article  MATH  Google Scholar 

  18. C. L. Prather and J. K. Shaw, Zeros of the successive derivatives of functions analytic in a neighborhood of a single pole, Michigan Math. J. 29 (1982), 111–119.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich IV / Mathematik, Universität Trier, D-54286, Trier, Germany

    Thierry Meyrath & Jürgen Müller

Authors
  1. Thierry Meyrath
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürgen Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thierry Meyrath.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Meyrath, T., Müller, J. On the behaviour of the successive derivatives of meromorphic functions on the final set. JAMA 120, 131–149 (2013). https://doi.org/10.1007/s11854-013-0017-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-013-0017-y

Keywords

  • Natural Number
  • Entire Function
  • Meromorphic Function
  • Principal Part
  • Successive Derivative