Some results about the size of the exceptional set in Nevanlinna's second fundamental theorem (Q1100310)

An essential tool in Nevanlinna's theory of meromorphic functions in the plane is an estimate for the proximity function of the logarithmic derivative. I.e. given any meromorphic function f one gets in principle (with \(0<r<R)\) \[ (1)\quad m(r,f'/f)=\frac{1}{2\pi}\int^{2\pi}_{0}\log \quad + | \frac{f'(re^{i\vartheta})}{f(re^{i\vartheta})}| d\vartheta \quad \leq \] \[ \leq \quad C_ 1 \log \quad + T(R,f)+C_ 2 \log \quad + (\frac{1}{R-r})\quad +\quad uninteresting\quad terms. \] Here T(r,f) denotes the Nevanlinna characteristic function [comp. e.g. \textit{R. Nevanlinna}, Eindeutige analytische Funktionen (1953; Zbl 0278.30002)], the reviewer and \textit{L. Volkmann}, Meromorphe Funktionen und Differentialgleichungen (1985)]. Using a lemma of Borel one can now derive from (1) an estimate \[ (2)\quad m(r,f'/f)\quad <\quad O(\log (rT(r,f))),\quad r\not\in E, \] where E (a set of exceptional values) can be covered by disjoint intervals of finite total length. The present paper gives two variations of Borels lemma to show that the weaker estimate \[ (3)\quad m(r,f'/f) = o(T(r,f)),\quad r\not\in E, \] is still valid in a larger set than (2). On the one hand the exceptional set E is characterized by \(\int_{E}r^{\lambda} dr<\infty\), for every \(\lambda >0\), E independent of \(\lambda\). And on the other hand E can be covered by disjoint intervals \([r_ n,r_ m+\delta_ m]\) such that \(\delta_ n<1/\Psi (n)\) 2, where \(\Psi (1)=1\), \(\Psi (n)=\exp (\Psi (n-1))\).