A new proof of Drasin's theorem on meromorphic functions of finite order with maximal deficiency sum. I
type: Item
instance of: scholarly article
title: A new proof of Drasin's theorem on meromorphic functions of finite order with maximal deficiency sum. I
zbMATH Open document ID: 0722.30018
DOI: 10.1007/BF01095413
author: Alexandre Eremenko
published in: Journal of Soviet Mathematics
publication date: 1990-01-01T00:00:00Z
review text: In 1929 F. Nevanlinna stated the following conjecture. Let f be a meromorphic function of finite order \(\rho\) with maximal deficiency sum: \[ \sum_{a\in {\bar {\mathbb{C}}}}\delta (a,f)=2. \] Then: 1) \(2\rho\) is a natural number \(\geq 2\). 2) If \(\delta (a)>0\), then \(\delta (a)=p(a)/\rho\), where p(a) is a natural number. 3) All the deficiency values are asymptotic. The complete proof of these statements has been obtained by \textit{D. Drasin} [Acta Math. 158, 1-94 (1987; Zbl 0622.30028)]. D. Drasin's proof was rather long and complicated. It makes use of a series of miscellaneous auxiliary tools. A. Eremenko found a new more accessible proof, based on the classical potential theory. Recently the method of the reviewed paper found new applications: the author and the reviewer proved ``a small function's'' version of the F. Nevanlinna conjecture in \textit{A. Eremenko} and \textit{M. Sodin} [Teor. Funkts. Funkts. Anal. Prilozh. (to appear)].
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 31A15
zbMATH DE Number: 4190099
zbMATH Keywords: maximal deficiency sum
reviewed by: M. I. Sodin
MaRDI profile type: Publication
cites work: Growth of Subharmonic Functions
cites work: On topologies and boundaries in potential theory. Enlarged ed. of a course of lectures delivered in 1966
description: scientific article; zbMATH DE number 4190099