On Nevanlinna's error terms
The main result of this paper is an estimate for the logarithmic derivative of meromorphic functions. The author shows that \[ m(r,(f'/f))\leq\log T(r,f)+\log \Psi(T(r,f))+(1/2)\log \Psi(r), \] where \(\Psi\) is a function for which \(\int_ 1^ \infty{1\over \Psi(x)}dx,\infty\) holds. If this condition does not hold he shows that there exists an entire function \(f\) such that the above estimate holds in the opposite direction. This means that the above result is sharp in a certain sense. Moreover he gives an estimate for the ramification term of the second fundamental theorem of Nevanlinna. The proof of the above estimate is based on results of \textit{S. Lang} [Lect. Notes Math. 1433, 174 p. (1990; Zbl 0709.30030)]. To prove the sharpness of his results the author uses a construction of an entire function from \textit{J. Miles} [J. Lond. Math. Soc., II. Ser. 45, No. 2, 243-254 (1992; Zbl 0706.30023)]. Reviewer's remark: A better estimate as above can be obtained using a result from \textit{A. A. Gol'dberg} and \textit{V. A. Grinshtejn} [Mat. Zametki 19, 525-530 (1976; Zbl 0337.30021)] combined with a lemma of Borel, see \textit{A. Hinkkanen} [Invent. Math. 108, 549-574 (1992; reviewed above)]. Then the last term \({1\over 2}\log \Psi(r)\) is the above estimate can be omitted. The obtained result is sharp as one can see with the construction of J. Miles.
scientific article; zbMATH DE number 58546