Journal d'Analyse Mathématique

, Volume 130, Issue 1, pp 71–89 | Cite as

The Schwarzian derivative and the Wiman-Valiron property

Article

Abstract

Let f be a transcendental meromorphic function in the plane such that f has finitely many critical values, the multiple points of f have bounded multiplicities, and the inverse function of f has finitely many transcendental singularities. Using the Wiman-Valiron method, it is shown that if the Schwarzian derivative Sf of f is transcendental, then f has infinitely many multiple points, the inverse function of Sf does not have a direct transcendental singularity over ∞, and ∞ is not a Borel exceptional value of Sf. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.

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References

  1. [1]
    W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamer. 11 (1995), 355–373.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Camb. Phil. Soc. (3) 142 (2007), 133–147.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    W. Bergweiler, P. J. Rippon, and G. M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, Proc. London Math. Soc. (3) 97 (2008), 368–400.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    D. Drasin, Proof of a conjecture of F. Nevanlinna concerning functions which have deficiency sum two, Acta. Math. 158 (1987), 1–94.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    G. Elfving, Über eine Klasse von Riemannschen Flächen und ihre Uniformisierung, Acta Soc. Sci. Fenn. 2 (1934), 1–60.MATHGoogle Scholar
  6. [6]
    A. Eremenko, Meromorphic functions with small ramification, Indiana Univ. Math. J. 42 (1994), 1193–1218.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    A. Eremenko, Geometric theory of meromorphic functions, In the Tradition of Ahlfors-Bers, III, Contemp. Math. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 221–230.CrossRefMATHGoogle Scholar
  8. [8]
    A. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier Grenoble 42 (1992), 989–1020.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    A. A. Gol’dberg and I. V. Ostrovskii, Value Distribution of Meromorphic Functions, Nauk, Moscow, 1970; English transl., Amer. Math. Soc. Providence, RI, 2008.Google Scholar
  10. [10]
    G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88–104.MathSciNetMATHGoogle Scholar
  11. [11]
    W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75–84.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  13. [13]
    W. K. Hayman, The local growth of power series: a survey of the Wiman-Valiron method, Canad. Math. Bull. 17 (1974), 317–358.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    W. K. Hayman, Multivalent Functions, 2nd edition, Cambridge University Press, Cambridge, 1994.CrossRefMATHGoogle Scholar
  15. [15]
    E. Hille, Lectures on Ordinary Differential Equations, Addison-Wesley, Reading, Mass., 1969.MATHGoogle Scholar
  16. [16]
    E. Hille, Ordinary Differential Equations in the Complex Domain, Wiley, New York, 1976.MATHGoogle Scholar
  17. [17]
    F. Iversen, Recherches sur les fonctions inverses des fonctions méromorphes, Thesis, University of Helsinki, Imprimerie de la Société de Littérature Finnoise, Helsinki, 1914.MATHGoogle Scholar
  18. [18]
    G. Jank and L. Volkmann, Einführung in die Theorie der ganzen und meromorphen Funktionen mit Anwendungen auf Differentialgleichungen, Birkhäuser, Basel, 1985.CrossRefMATHGoogle Scholar
  19. [19]
    J. K. Langley, Proof of a conjecture of Hayman concerning f and f, J. London Math. Soc. (2) 48 (1993), 500–514.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Z. Nehari, Conformal Mapping, Dover, New York, 1975.MATHGoogle Scholar
  22. [22]
    R. Nevanlinna, Über eine Klasse meromorpher Funktionen, Septieme Congres Math. Scand. Jbuch. 56 (Oslo, 1929), 81–83.Google Scholar
  23. [23]
    R. Nevanlinna, Über Riemannsche Flächen mit endlich vielen Windungspunkten, Acta Math. 58 (1932), 295–373.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    R. Nevanlinna, Eindeutige analytische Funktionen, 2. Auflage, Springer, Berlin, 1953.CrossRefMATHGoogle Scholar

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© Hebrew University Magnes Press 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamEngland

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