Relatively (R)-dense universal sequences for certain classes of functions

Abstract

Let \(f:\mathbb{R} ^+\rightarrow \mathbb{R} ^+\) and \((a_n)^\infty_{n=1}\) be a sequence of positive reals. We will say that \((a_n)^\infty_{n=1}\) is relatively \((R)\)-dense for \(f\) provided that for every \(x,y\in \mathbb{R} ^+\) with \(f(x)\lt f(y)\) there exists \(n,m\in\mathbb{N} \) such that \(f(x)\lt \frac{f(a_n)}{f(a_m)}\lt f(y)\). Sufficient conditions are given for a sequence of positive reals to be relatively \((R)\)-dense for certain functions.