Journal d’Analyse Mathématique

, Volume 71, Issue 1, pp 87–102 | Cite as

Sharp forms of nevanlinna’s error terms

Article

Abstract

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ 1 dt/ψ(t) = ∞ and ∫ 1 dt/p(t) = ∞, then\(S(r,f) \le \log + \frac{{\psi \left( {T(r,f)} \right)}}{{p(r)}} + O(1)\) asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ 1 r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible.

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References

  1. [1]
    A. A. Gol’dberg and V. A. Grinshtein,The logarithmic derivative of a merornorphic function (Russian), Mat. Zametki19 (1976), 525–530.MATHMathSciNetGoogle Scholar
  2. [2]
    W. K. Hayman,Meromorphic Functions, Clarendon Press, Oxford, 1964.MATHGoogle Scholar
  3. [3]
    A. Hinkkanen,A sharp form of Nevanlinna’s second fundamental theorem, Invent. Math.108 (1992), 549–574.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    M. Jankowski,An estimate for the logarithmic derivative of meromorphic functions, Analysis14 (1994), 185–195.MATHMathSciNetGoogle Scholar
  5. [5]
    S. Lang,The error term in Nevanlinna theory, Duke Math. J.56 (1988), 193–218.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    S. Lang and W. Cherry,Topics in Nevanlinna theory, Lecture Notes in Math., Vol.1443, Springer, Berlin, 1990.Google Scholar
  7. [7]
    J. Miles,A sharp form of the lemma on the logarithemic derivative, J. London Math. Soc.45 (2) (1992), 243–254.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Nevanlinna,Analytic Functions, Princeton University Press, Princeton, 1960.MATHGoogle Scholar
  9. [9]
    C. F. Osgood,Sometimes effective ThueSiegel-Schmidt-Nevanlinna bounds, or better, J. Number Theory21 (1985), 347–389.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    P. Vojta,Diophantine approximations and value distribution theory, Lecture Notes in Math., Vol. 1239, Springer, Berlin, 1987.Google Scholar
  11. [11]
    P. Wong,On the second main theorem of Nevanlinna theory, Amer. J. Math.111 (1989), 549–583.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    L. Yang,Value Distribution Theory, Springer-Verlag and Science Press, Berlin, 1993.MATHGoogle Scholar
  13. [13]
    Z. Ye,On Nevanlinna’s error terms, Duke Math. J.64 (1991), 243–260.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 1997

Authors and Affiliations

  1. 1.Fachbereich 3 MathematikTechnische UniversitÄt BerlinBerlinGermany
  2. 2.Institute of MathematicsAcademia SinicaBeijingChina

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