On the frequency of multiple values of a meromorphic function of small order
type: Item
instance of: scholarly article
title: On the frequency of multiple values of a meromorphic function of small order
zbMATH Open document ID: 0573.30034
author: Daniel F. Shea
published in: Michigan Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: Let f be meromorphic in the plane, of (lower) order \(\mu\), and let \(N_ 1(r)\) be the usual branching term. The main result gives a lower asymptotic bound for the ratio \(N_ 1(r)/T(r,F)\) when \(\mu <\). As an application, it is shown that if g and h are linearly independent transcendental functions of order \(<\), then the Wronskian \(W(g,h)=gh'- hg'\) must always have approximately as many zeros as allowed by considering the growths of f and g. Such behavior is conjectured to persist whenever \(2\mu\) is not an integer \(\geq 2\), but this has proved intractible. The proof here uses the hypotheses to construct regions which surround the origin, in which \(| f|\) is large, and then uses the relation between f' and W(g,h) when \(f=g/h\). There is also significant use of a theory of Pólya peaks due to him with \textit{A. E. Eremenko} and \textit{M. L. Sodin} [Minimum modulus theorems for entire functions at Pólya peaks, Teor. Funkts., Funkts. Anal. Prilozh. (to appear)].
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 3914678
reviewed by: David Drasin
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003137
OpenAlex ID: W2068848350
DOI: 10.1307/MMJ/1029003137
description: scientific article; zbMATH DE number 3914678