Abstract
Let f be a transcendental meromorphic function, T(r,f) its characteristic function and S(r, f) the error term in Nevanlinna's second fundamental theorem. It is shown that for every increasing function ψ(r) such that log ψ(r) = o(r) we have S(r, f) = o(T(r, f)) outside an exceptional set E satisfying ∫ E ψ(T(r, f))dr < ∞. This result makes clear the relationship between the size of the exceptional set and the growth of the characteristic function and implies that for functions of rapid lower growth improved conditions on the size of the exceptional set can be given. A general example of an entire function with a suitable exceptional set is constructed, showing that these results are essentially best possible.
Similar content being viewed by others
REFERENCES
A. FernÁndez, Some results about the size of the exceptional set in Nevanlinna's second fundamental theorem, Collect. Math. 37 (1986), 229-238.
A. FernÁndez, On the size of the exceptional set in Nevanlinna theory, J. London Math. Soc. (2) 34 (1986), 449-456.
W. K. Hayman, Meromorphic functions (Clarendon Press, Oxford, 1964).
W. K. Hayman, Die Nevanlinna-Characteristik von meromorphen Funktionen und ihrer Integralen. In Festband zum 70. Geburtstag von Rolf Nevanlinna, pp. 16-20, Springer, Berlin, 1966.
A. Hinkkanen, A sharp form of Nevanlinna's second fundamental theorem, Invent. Math. 108 (1992), 549-574.
S. Lang and W. Cherry, Topics in Nevanlinna theory, Lecture Notes in Math. 1433, Springer, Berlin, 1990.
J. Miles, A sharp form of the lemma on the logarithmic derivative, J. London Math. Soc. (2) 45 (1992), 243-254.
R. Nevanlinna, Analytic functions, Springer-Verlag, Berlin, 1970.
F. RodrÍguez, On the size of the exceptional set in Nevanlinna's second fundamental theorem for certain classes of meromorphic functions, Math. Nachr. 178 (1996), 247-258.
Z. Ye, On Nevanlinna's error terms, Duke Math. J. 64 (1991), 243-260.
Rights and permissions
About this article
Cite this article
Mateos, F.R. Growth and Size of the Exceptional Set in Nevanlinna's Second Fundamental Theorem. Periodica Mathematica Hungarica 35, 199–210 (1997). https://doi.org/10.1023/A:1004505501177
Issue Date:
DOI: https://doi.org/10.1023/A:1004505501177