\({\mathcal O}\)-regularly varying functions in approximation theory
A positive Lebesgue measurable function \(\varphi: (0,1] \to\mathbb{R}\) is said to be O-regularly varying at zero if \(\varphi\) and \(1/ \varphi\) are bounded on every interval \((t_0,1]\) for each \(t_0\in (0,1)\), and \[ 0<\lim \inf(t \to 0+)\quad \varphi(Ct)/ \varphi(t),\;\lim \sup (t\to 0+)\quad\varphi (Ct)/ \varphi(t) <\infty \] for each \(C\in (0,1]\). (This definition is due to \textit{S. Aljančić} and \textit{D. Aranđelović} [Publ. Inst. Math. Beograd, Nouv. Sér. 22(36), 5-22 (1977; MR 57 \# 6317)]). For positive functions \(f_1,f_2: (0,1] \to\mathbb{R}\), the \textit{growth relation} \(f_1<f_2\) is said to hold if there is a constant \(C\in (0,1)\) such that \(\lim \sup (t\to 0+)\) \(\{f_1(Ct) f_2(Ct)\}/ \{f_1 (t) f_2(Ct)\} <1\). The relation is explored in terms of the ``Matuszewska'' indices when \(f_1\) and \(f_2\) are taken as O-regularly varying, and is used to characterize a third function \(\chi(t)\) as O-regularly varying, motivated by some applications in approximation theory.
scientific article; zbMATH DE number 1139489