Skip to main content
SpringerLink
Log in

An Extension of a Normality Result of D. Drasin and H. Chen & X. Hua for Analytic Functions

  • Published:
Computational Methods and Function Theory Aims and scope Submit manuscript

Abstract

We show that if an entire function f satisfies

$$af^n(z) + f^{(k)}(z) + P\lbrack f \rbrack(z) \ne 0$$

for all z ∈ C, for some n ≥ 2, k ≥ 1, a ≠ 0, and with P a differential polynomial of a certain form, then f must be a constant. We also prove the corresponding normality criterion where the coefficients are meromorphic functions. This generalizes results of Hayman [7], Drasin [5] and Chen and Hua [2].

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Chen and Y. Gu, An improvement of Marty’s criterion and its applications, Sci. Sinica Ser. A 36 (1993), 674–681.

    MathSciNet  MATH  Google Scholar 

  2. H. Chen and X. Hua, Normal families of holomorphic functions, J. Austral. Math. Soc., Ser. A 59 (1995), 112–117.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Clunie, On integral and meromorphic functions, J. Lond. Math. Soc. 37 (1962), 17–27.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Döringer, Exceptional values of differential polynomials, Pacific J. Math. 98 (1982), 55–62.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Drasin, Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231–263.

    Article  MathSciNet  MATH  Google Scholar 

  6. W. K. Hayman, Meromorphic Functions, Oxford University Press, London, 1964.

    MATH  Google Scholar 

  7. W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math. 70 2 (1959), 9–42.

    Article  MathSciNet  MATH  Google Scholar 

  8. Xiu-Hou Hua, Some extensions of the Tumura-Clunie Theorem, Compl. Var. 16 (1991), 69–77.

    Article  MATH  Google Scholar 

  9. J. Langley, On normal families and a result of Drasin, Proc. Roy. Soc. Edin. 98 A (1984), 385–393.

    Article  MathSciNet  Google Scholar 

  10. E. Mues, Über ein Problem von Hayman, Math. Z. 164 (1979), 239–259.

    Article  MathSciNet  MATH  Google Scholar 

  11. X. Pang, On normal criterion of meromorphic functions, Sci. Sinica 33 5 (1990), 521–527.

    MATH  Google Scholar 

  12. X. Pang and L. Zalcman, Normal families and shared values, Bull. Lond. Math. Soc. 32 3 (2000), 325–331.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Schiff, Normal Families, Springer, New York, 1993.

    Book  MATH  Google Scholar 

  14. Y. Tumura, On the extensions of Borel’s theorem and Saxer-Csillag’s theorem, Proc. Phys. Math. Soc. Japan 19 3 (1937), 29–35.

    Google Scholar 

  15. L. Zalcman, New light on normal families, in: Proceedings of the Ashkelon Workshop on Complex Function Theory (May 1996), IMCP, Vol. 11, Amer. Math. Soc. 1997, 237–245.

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Jüurgen Grahl

Authors
  1. Jüurgen Grahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jüurgen Grahl.

Additional information

Part of this work was supported by GIF (No. G 43–117.6/1999) and INTAS (No. INTAS 99–00089)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grahl, J. An Extension of a Normality Result of D. Drasin and H. Chen & X. Hua for Analytic Functions. Comput. Methods Funct. Theory 1, 457–478 (2001). https://doi.org/10.1007/BF03321002

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03321002

Keywords

2000 MSC

Search

Navigation