Fast growing solutions of linear differential equations in the unit disc
type: Item
instance of: scholarly article
title: Fast growing solutions of linear differential equations in the unit disc
zbMATH Open document ID: 1120.34071
author: Janne Heittokangas
author: Risto Korhonen
author: Jouni Rättyä
published in: Results in Mathematics
publication date: 2007-02-15T00:00:00Z
review text: In several recent papers, the authors have studied solutions of complex differential equations in the unit disc \(\mathbb{D}\), see e.g.\ [Ann. Acad. Sci. Fenn. Math. Diss. 122, 1--54 (2000; Zbl 0965.34075), Ann. Acad. Sci. Fenn. Math. 29, 233--246 (2004; Zbl 1057.34111), Proc. Am. Math. Soc. 135, No. 5, 1355--1363 (2007; Zbl 1122.34067)], making use of notions from complex function spaces to control solutions that have their growth comparable to that of the coefficients. This paper is devoted to considering fast growing solutions, in which case one may expect to obtain results similar to those valid in the whole plane. The authors here make use of iterated logarithms to define the \(n\)-order of an analytic function \(f\) as \[ \sigma_{M,n}(f):=\limsup_{r\to1^-}\frac{\log^+_{n-1}M(r,f)}{-\log(1-r)}; \] here, \(\log^+_1x:=\max(\log x,0)\) and \(\log_{n+1}^+x=\log^+\log^+_nx\). Several results are now proved, together with a number of examples showing the sharpness of the results obtained. As an example of the results, consider solutions \(f\) of \[ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\dots+a_1(z)f'+a_0(z)f=0\tag{*} \] with coefficients analytic in~\(\mathbb{D}\). Then \(\sigma_{M,n+1}(f)\leq\alpha\) if and only if \(\sigma_{M,n}(a_j)\leq\alpha\) holds for all \(j=0,\dots,k-1\). Moreover, if \(q\) is the largest index for which \(\sigma_{M,n}(a_q)=\max_{0\leq j\leq k-1}\sigma_{M,n}(a_j)\), then there are at least \(k-q\) linearly independent solutions \(f\) of~(*) such that \(\sigma_{M,n+1}(f)=\sigma_{M,n}(a_q)\). The proofs in this paper strongly rely on logarithmic derivative estimates in the unit disc, see the estimates by the first author, \textit{G. G. Gundersen} and \textit{I. Chyzhykov} [Proc. London Math. Soc. 86, No. 3, 735--754 (2003; Zbl 1044.34049)] and their refinements in this paper. We finally remark that a closely related, independent, paper by \textit{T. Cao} and \textit{H. Yi} [J. Math. Anal. Appl. 319, 278--294 (2006; Zbl 1105.34059)] contains similar, and…
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5127413
MaRDI profile type: Publication
full work available at URL: s00025-006-0223-3
OpenAlex ID: W2170003428
DOI: 10.1007/S00025-006-0223-3
description: scientific article; zbMATH DE number 5127413