Illinois Journal of Mathematics

Nonvanishing derivatives and the MacLane class $\mathcal{A}$

Alastair Fletcher, Jim Langley, and Janis Meyer

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Abstract

Let k≥2 and let f be meromorphic in the unit disc Δ, such that f(z)f(k)(z)≠0 for all z∈Δ and the poles of f in Δ have bounded multiplicities. Then f has asymptotic values on a dense subset of Δ.

Article information

Source
Illinois J. Math. Volume 53, Number 2 (2009), 379-390.

Dates
First available in Project Euclid: 23 February 2010

Permanent link to this document
http://projecteuclid.org/euclid.ijm/1266934783

Digital Object Identifier
doi:10.1214/09-AAP195

Zentralblatt MATH identifier
05676327

Mathematical Reviews number (MathSciNet)
MR2594634

Subjects
Primary: 30D40: Cluster sets, prime ends, boundary behavior 30D35: Distribution of values, Nevanlinna theory

Citation

Fletcher, Alastair; Langley, Jim; Meyer, Janis. Nonvanishing derivatives and the MacLane class $\mathcal{A}$ . Illinois J. Math. 53 (2009), no. 2, 379--390. doi:10.1214/09-AAP195. http://projecteuclid.org/euclid.ijm/1266934783.


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References

  • K. F. Barth, Asymptotic values of meromorphic functions, Michigan Math. J. 13 (1966), 321--340.
  • K. F. Barth and P. J. Rippon, Exceptional values and the Maclane class $\mathcalA$, Bergman spaces and related topics in analysis, Contemp. Math. 404 (2006), 41--52.
  • W. Bergweiler and J. K. Langley, Nonvanishing derivatives and normal families, J. Analyse Math. 91 (2003), 353--367.
  • J. G. Clunie, On integral and meromorphic functions, J. London Math. Soc. 37 (1962), 17--27.
  • G. Frank, Eine Vermutung von Hayman über Nullstellen meromorpher Funktionen, Math. Zeit. 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 und deren Ableitungen, Math. Ann. 225 (1977), 145--154.
  • G. Frank and J. K. Langley, Pairs of linear differential polynomials, Analysis 19 (1999), 173--194.
  • W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. of Math. 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 Studies in Math., vol. 15, Walter de Gruyter, Berlin, 1993.
  • J. K. Langley, Proof of a conjecture of Hayman concerning $f$ and $f'' $, J. London Math. Soc. (2) 48 (1993), 500--514.
  • G. R. Maclane, Asymptotic values of holomorphic functions, Rice Univ. Studies 49 (1963), 83 pp.
  • G. R. Maclane, Exceptional values of $f^(n)(z)$, asymptotic values of $f(z)$, and linearly accessible asymptotic values, Mathematical essays dedicated to A.J. Macintyre 271--288, Ohio Univ. Press, 1970.
  • G. Pólya, Über die Nullstellen sukzessiver Derivierten, Math. Zeit. 12 (1922), 36--60.
  • W. Schwick, Normality criteria for families of meromorphic functions, J. Analyse Math. 52 (1989), 241--289.
  • Y. Tumura, On the extensions of Borel's theorem and Saxer--Csillag's theorem, Proc. Phys. Math. Soc. Japan (3) 19 (1937), 29--35.
  • L. Zalcman, A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813--817.
  • L. Zalcman, Normal families: New perspectives, Bull. Amer. Math. Soc. (N.S.) 35 (1998), 215--230.