Journal d’Analyse Mathématique

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

Nonvanishing derivatives and normal families

  • Walter Bergweiler
  • J. K. Langley


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.


Rational Function Entire Function Meromorphic Function Normal Family Simple Pole 
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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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