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Journal d’Analyse Mathématique

, Volume 91, Issue 1, pp 353–367 | Cite as

Nonvanishing derivatives and normal families

  • Walter Bergweiler
  • J. K. Langley
Article

Abstract

We consider the differential operators Ψ k , defined by Ψ1(y) =y and Ψ k+1(y)=yΨ k y+d/dz k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z 2+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ k (f /f) =f (k)/f, we deduce in particular that iff andf (k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f /f :fF} is normal.

Keywords

Rational Function Entire Function Meromorphic Function Normal Family Simple Pole 
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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References

  1. [1]
    W. Bergweiler,Normality and exceptional values of derivatives, Proc. Amer. Math. Soc.129 (2001), 121–129.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    J. Clunie,On integral and meromorphic functions, J. London Math. Soc.37 (1962), 17–27.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Eremenko,Meromorphic functions with small ramification, Indiana Univ. Math. J42 (1994), 1193–1218.CrossRefMathSciNetGoogle Scholar
  4. [4]
    G. Frank,Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z.149 (1976), 29–36.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    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.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    G. Frank, W. Hennekemper and G. Polloczek,Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen, Math. Ann.225 (1977), 145–154.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    G. Frank and J. K. Langley,Pairs of linear differential polynomials, Analysis19 (1999), 173–194.MATHMathSciNetGoogle Scholar
  8. [8]
    W. K. Hayman,Picard values of meromorphic functions and their derivatives, Ann. Math. (2)70 (1959), 9–42.MathSciNetGoogle Scholar
  9. [9]
    W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  10. [10]
    E. L. Ince,Ordinary Differential Equations, Dover, New York, 1956.Google Scholar
  11. [11]
    I. Laine,Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin/New York, 1993.Google Scholar
  12. [12]
    J. K. Langley,Proof of a conjecture of Hayman concening f and f″, J. London Math. Soc. (2)48 (1993), 500–514.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    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.MATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    Xuecheng Pang,Shared values and normal families, Analysis22, (2002), 175–182.MATHGoogle Scholar
  15. [15]
    Xuecheng Pang and L. Zalcman,Normal families and shared values, Bull. London Math. Soc.32 (2000), 325–331.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    J. Schiff,Normal Families, Springer, New York, Berlin, Heidelberg, 1993.MATHGoogle Scholar
  17. [17]
    W. Schwick,Normality criteria for families of meromorphic functions, J. Analyse Math.52 (1989), 241–289.MATHMathSciNetGoogle Scholar
  18. [18]
    D. Shea,On the frequency of multiple values of a meromorphic function of small order, Michigan Math. J.32 (1985), 109–116.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    L. Zalcman,Normal families: new perspectives, Bull. Amer. Math. Soc., N.S.35 (1998), 215–230.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2003

Authors and Affiliations

  • Walter Bergweiler
    • 1
  • J. K. Langley
    • 2
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.School of Mathematical SciencesUniversity of NottinghamUK

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