\(\mathcal O\)-regularly varying functions and strong asymptotic equivalence (Q1270842)

A positive measurable function \(f\) on \([\alpha,\infty)\) such that \(f(x(t))\sim f(y(t))\) as \(t\to\infty\), for any positive functions \(x(t)\), \(y(t)\) on \([\alpha,\infty)\) which have the properties \(x(t)\to\infty\), \(y(t)\to\infty\) and \(x(t)\sim y(t)\) as \(t\to\infty\) is said to be CRV. The author shows (Characterization Theorem) that \(f\) is CRV if and only if its index function \(f_k(\lambda)= \limsup(x\to\infty)\;f(\lambda x)/f(x)\) is continuous in \(\lambda>0\). The CRV functions are a generalization of the regularly varying functions (RVF) and are a subset of the O-regularly varying functions (ORV) which merely require finiteness of the index function for each \(\lambda>0\) [\textit{S. Aljančić} and \textit{D. Arandjelović}, Publ. Inst. Math., Beograd, N. Ser. 22, 5-22 (1977; Zbl 0379.26003)]. Accordingly, in the tradition of regular function theory, an integral representation theorem is also given.