Karamata functions and differential equations: achievements from the 20th century
type: Item
instance of: scholarly article
title: Karamata functions and differential equations: achievements from the 20th century
zbMATH Open document ID: 1429.34003
author: Aleksandar Nikolić
published in: Historia Mathematica
publication date: 2018-09-07T00:00:00Z
review text: In his seminal article [Mathematica, Cluj 4, 38--53 (1930; JFM 56.0907.01)], \textit{J. Karamata} studied positive functions \(r\) on intervals \([a,+\infty)\), \(a>0\), such that for some exponent \(\rho\), one has \(\lim_{x\to\infty}r(tx)/r(x)=t^\rho\) for all \(t>0\), with the aim of generalising one of the Hardy-Littlewood Tauberian theorems (see [\textit{J. Korevaar}, Tauberian theory. A century of developments. Berlin: Springer (2004; Zbl 1056.40002)]). These functions are called today \textit{Karamata\/} or \textit{regularly varying\/} functions with \textit{regularity index~\(\rho\)}. There also is the opposite concept of \textit{rapidly varying\/} functions for which the limit is either~\(0\) or~\(\infty\) for all~\(t>1\); there are generalisations and variants: see, e.g. [\textit{V. G. Avakumovic}, Bull. Int. Acad. Yougoslave Sci. Beaux-Arts 29/30, 107--117 (1936; Zbl 0015.25003); \textit{E. Omey} and \textit{E. Willekens}, J. Lond. Math. Soc., II. Ser. 37, No. 1, 105--118 (1988; Zbl 0612.26001); \textit{J. Jaroš} and \textit{T. Kusano}, Bull., Cl. Sci. Math. Nat., Sci. Math. 129, No. 29, 25--60 (2004; Zbl 1091.34521)]. A general reference is [\textit{N. H. Bingham} et al., Regular variation. Cambridge: Cambridge University Press (1987; Zbl 0617.26001)]. The author describes the rôle of these classes of functions in the subject of differential equations through a historical survey; see also [\textit{V. Marić}, Regular variation and differential equations. Berlin: Springer (2000; Zbl 0946.34001)]. In \S2, the author focuses on nonlinear differential equations: \textit{V. G. Avakumović} [Acad. Serbe, Bull. Acad. Sci. Math. Natur. A 1, 101--113 (1947); ibid 2, 223--235 (1948; Zbl 0033.27302)] studied the asymptotics of solutions of the following Thomas-Fermi-type equation: \(y''=f(x)y^\lambda\) with boundary conditions \(y(0)=1\) and \(\lim_{x\to\infty}y(x)=0\), where \(f\)~is equivalent to a Karamata function of regularity index~\(\rho>-2\). This thread of r…
reviewed by: Stefan Neuwirth
Mathematics Subject Classification ID: 34-03
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 6934129
zbMATH Keywords: Karamata function
zbMATH Keywords: regularly varying function
zbMATH Keywords: Thomas-Fermi equation
zbMATH Keywords: Friedmann equation
MaRDI profile type: Publication
full work available at URL: j.hm.2017.12.001
OpenAlex ID: W2793548273
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Karamata functions and differential equations: achievements from the 20th century (DOI)
DOI: 10.1016/J.HM.2017.12.001