Value distribution for a class of small functions in the unit disk
type: Item
instance of: scholarly article
title: Value distribution for a class of small functions in the unit disk
zbMATH Open document ID: 1216.30034
DOI: 10.1155/2011/537478
author: Paul A. Gunsul
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2011-05-27T00:00:00Z
full work available at URL: https://eudml.org/doc/224790
review text: Summary: If \(f\) is a meromorphic function in the complex plane, R. Nevanlinna noted that its characteristic function \(T(r,f)\) can be used to categorize \(f\) according to its rate of growth as \(|z|=r\to \infty\). Later, H. Milloux showed for transcendental meromorphic functions in the plane that, for each positive integer \(k\), \[ m\bigg(r,\frac{f^{(k)}}{f}\bigg)= o\big(T(r,f)\big)\quad\text{as}\quad r\to \infty, \] possibly outside a set of finite measure, where \(m\) denotes the proximity function of Nevanlinna theory. If \(f\) is a meromorphic function in the unit disk \(D=\{ z:| z|<1\}\), analogous results to the previous equation exist when \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}=+\infty. \] In this paper, we consider the class of meromorphic functions \(\mathcal P\) in \(D\) for which \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}<+\infty,\quad \limsup_{r\to 1^-}T(r,f)=+\infty \] and \[ m\bigg(r,\frac{f}{f'}\bigg)=o\big(T(r,f)\big)\quad \text{as}\quad r\to 1. \] We explore the characteristics of this class and some places where functions in the class behave in a significantly different manner than those for which \[ \limsup_{r\to 1^-}\frac{T(r,f)}{\log\frac{1}{1-r}}=+\infty \] holds. We also explore connections between the class \(\mathcal P\) and linear differential equations and values of differential polynomials, and give an analogue to Nevanlinna's five-value theorem.
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5900730
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: meromorphic function
Wikidata QID: Q58687851
MaRDI profile type: Publication
OpenAlex ID: W2083538045
cites work: Q3754247
cites work: On the growth of the logarithmic derivative
cites work: Q4084504
cites work: Unbounded functions in the unit disc
cites work: Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class
cites work: Functions analytic in a finite disk and having asymptotically prescribed characteristic
cites work: Linear differential equations with coefficients in weighted Bergman and Hardy spaces
cites work: Finite order solutions of linear differential equations in the unit disc
cites work: Q3645663
cites work: On the characteristic of functions meromorphic in the unit disk and of their integrals
description: scientific article; zbMATH DE number 5900730