The Schwarzian derivative and the Wiman-Valiron property

Article, Preprint English OPEN
Langley, James (2016)
  • Publisher: Springer
  • Related identifiers: doi: 10.1007/s11854-016-0029-5
  • Subject: 30D35 | Mathematics - Complex Variables
    arxiv: Mathematics::Metric Geometry | Mathematics::Complex Variables

Consider a transcendental meromorphic function in the plane with finitely many critical values, such that the multiple points have bounded multiplicities and the inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that if the Schwarzian derivative is transcendental then the function has infinitely many multiple points, the inverse function does not have a direct transcendental singularity over infinity, and infinity is not a Borel exceptional value. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method.
  • References (24)
    24 references, page 1 of 3

    [1] W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of nite order, Rev. Mat. Iberoamericana 11 (1995), 355-373.

    [2] W. Bergweiler and J.K. Langley, Zeros of di erences of meromorphic functions, Math. Proc. Camb. Phil. Soc. 142 (2007), 133-147.

    [3] W. Bergweiler, P.J. Rippon and G.M. Stallard, Dynamics of meromorphic functions with direct or logarithmic singularities, , Proc. London Math. Soc. 97 (2008), 368-400.

    [4] D. Drasin, Proof of a conjecture of F. Nevanlinna concerning functions which have de ciency sum two, Acta. Math. 158 (1987), 1-94.

    [5] G. Elfving, U ber eine Klasse von Riemannschen Flachen und ihre Uniformisierung, Acta Soc. Sci. Fenn. 2 (1934) 1-60.

    [6] A. Eremenko, Meromorphic functions with small rami cation, Indiana Univ. Math. J. 42 (1994), 1193-1218.

    [7] A. Eremenko, Geometric theory of meromorphic functions, In the Tradition of Ahlfors-Bers, III, Contemp. Math. 355, (Amer. Math. Soc. Providence, 2004) 221-230.

    [8] A. Eremenko and M.Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier Grenoble 42 (1992), 989-1020.

    [9] A.A. Gol'dberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, (Nauka, Moscow, 1970 (Russian)). English transl., Translations of Mathematical Monographs 236, (Amer. Math. Soc. Providence 2008).

    [10] G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), 88-104.

  • Similar Research Results (2)
  • Metrics
    No metrics available
Share - Bookmark