Some results about the size of the exceptional set in Nevanlinna's second fundamental theorem
type: Item
instance of: scholarly article
title: Some results about the size of the exceptional set in Nevanlinna's second fundamental theorem
zbMATH Open document ID: 0639.30029
published in: Collectanea Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/38707
review text: 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 \(\Psi (1)=1\), \(\Psi (n)=\exp (\Psi (n-1))\).
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 4042252
author: Arturo Fernández Arias
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4042252