Abstract
We consider the differential operators Ψ k , defined by Ψ1(y) =y and Ψ k+1(y)=yΨ k y+d/dz(Ψ 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 :f ∈F} is normal.
Similar content being viewed by others
References
W. Bergweiler,Normality and exceptional values of derivatives, Proc. Amer. Math. Soc.129 (2001), 121–129.
J. Clunie,On integral and meromorphic functions, J. London Math. Soc.37 (1962), 17–27.
A. Eremenko,Meromorphic functions with small ramification, Indiana Univ. Math. J42 (1994), 1193–1218.
G. Frank,Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Z.149 (1976), 29–36.
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.
G. Frank, W. Hennekemper and G. Polloczek,Über die Nullstellen meromorpher Funktionen and ihrer Ableitungen, Math. Ann.225 (1977), 145–154.
G. Frank and J. K. Langley,Pairs of linear differential polynomials, Analysis19 (1999), 173–194.
W. K. Hayman,Picard values of meromorphic functions and their derivatives, Ann. Math. (2)70 (1959), 9–42.
W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964.
E. L. Ince,Ordinary Differential Equations, Dover, New York, 1956.
I. Laine,Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin/New York, 1993.
J. K. Langley,Proof of a conjecture of Hayman concening f and f″, J. London Math. Soc. (2)48 (1993), 500–514.
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.
Xuecheng Pang,Shared values and normal families, Analysis22, (2002), 175–182.
Xuecheng Pang and L. Zalcman,Normal families and shared values, Bull. London Math. Soc.32 (2000), 325–331.
J. Schiff,Normal Families, Springer, New York, Berlin, Heidelberg, 1993.
W. Schwick,Normality criteria for families of meromorphic functions, J. Analyse Math.52 (1989), 241–289.
D. Shea,On the frequency of multiple values of a meromorphic function of small order, Michigan Math. J.32 (1985), 109–116.
L. Zalcman,A heuristic principle in complex function theory, Amer. Math. Monthly82 (1975), 813–817.
L. Zalcman,Normal families: new perspectives, Bull. Amer. Math. Soc., N.S.35 (1998), 215–230.
Author information
Authors and Affiliations
Additional information
The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.
Rights and permissions
About this article
Cite this article
Bergweiler, W., Langley, J.K. Nonvanishing derivatives and normal families. J. Anal. Math. 91, 353–367 (2003). https://doi.org/10.1007/BF02788794
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02788794