ODEs and Wiman-Valiron theory in the unit disc
type: Item
instance of: scholarly article
title: ODEs and Wiman-Valiron theory in the unit disc
zbMATH Open document ID: 1205.30028
author: P. C. Fenton
author: John Rossi
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-03-29T00:00:00Z
review text: The order of an analytic function \(f\) in the unit disc is defined by \[ \rho:=\limsup_{r\to 1^{-}}\frac{\log\log M(r,f)}{-\log(1-r)}. \] The authors obtain Wiman-Valiron type theorems: Theorem. Suppose that \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\) is analytic in the unit disc, of order \(\rho>0\). Given \(\gamma\) satisfying \[ 0<\gamma<\frac{\rho}{2(\rho+1)}, \] let \(\zeta\) be such that \[ |f(\zeta)|\geq N^{-\gamma}M\big(|\zeta|, f\big), \] where \(N=N\big(|\zeta|,f\big)\). Let \(T_{j}\) be a sequence satisfying \[ \lim_{j\to\infty}T_{j}=1\quad\text{and}\quad \lim_{j\to\infty}\frac{\log N(T_{j}, f)}{-\log(1-T_{j})}=\rho+1. \] Then, for every positive integer \(q\), \[ f^{(q)}(\zeta)=\big(1+o(1)\big)\left(\frac{N}{\zeta}\right)^{q}f(\zeta) \] as \(|\zeta|\to 1^-\) outside a set \(E\) such that \[ \lim_{j\to \infty}\frac{m\big(E\cap(T_{j}, 1)\big)}{1-T_{j}}=0. \] Theorem. Suppose that \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\) is analytic in the unit disc, of order zero, and that \(\gamma:(0,1)\to\mathbb{R}\) is positive such that \(\gamma(t)\to 0\) as \(t\to 1^{-}\). If \(\big|f(\zeta)\big|\geq N^{-\gamma(|\zeta|)}M\big(|\zeta|, f\big)\), then for every positive integer \(q\) and any positive number \(\eta\), \[ \frac{f^{(q)}(\zeta)}{f(\zeta)}=O\left(\frac{1}{1-|\zeta|}\right)^{q+\eta} \] as \(|\zeta|\to 1^-\) outside a set of zero final density. The authors also apply the above theorems to ODEs with analytic coefficients in the unit disc.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 34M10
zbMATH DE Number: 5687687
zbMATH Keywords: power series
zbMATH Keywords: Wiman-Valiron theory
zbMATH Keywords: analytic function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.12.044
OpenAlex ID: W2021379705
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DOI: 10.1016/J.JMAA.2009.12.044
description: scientific article; zbMATH DE number 5687687