Weak Asymptotic Equivalence Relation and Inverse Functions in the Class OR

Dragan Đurčić, Aleksandar Torgašev

If $f(x)$ is a continuous, strictly increasing and unbounded function defined on an interval $[a,+\infty)$, $(a>0)$, in this paper we shall prove that $f^{-1}(x)$, $(x\geq a)$ belongs to the Karamata class OR of all $\mathcal{O}$-regularly varying functions, if and only if for every function $g(x)$, $(x\geq a)$, which satisfies $f(x) \asymp g(x)$ as $x\to\infty$, we have $f^{-1}(x) \asymp g^{-1}(x)$ as $x\to +\infty$. Here, $\asymp$ is the weak asymptotic equivalence relation. We shall also prove some variants of the previous theorem, in which, except the weak, we also deal with the strong asymptotic equivalence relation.