Relatively \((R)\)-dense universal sequences for certain classes of functions
A sequence \((a_n)^\infty_{n = 1}\), \(a_n \in R^+\) \((n = 1,2, \dots)\) is said to be a relatively \((R)\)-dense universal sequence for a class \(M\) of functions \(f:R^+ \to R^+\) provided that for each \(f \in M\) the following holds: If \(x,y \in R^+\) and \(f(x) < f(y)\), then there are \(m,n\) such that \[ f(x) < {f(a_n) \over f(a_m)} < f(y). \] Denote by \(F\) the class of all \(f : R^+ \to R^+\) satisfying the conditions: 1) \(f(xy) \geq f(x) f(y)\) \((x,y \in R^+)\) 2) \(f\) is continuous at 1 and \(f(1) = 1\). Then a sequence \((a_n)^\infty_1\) is a relatively \((R)\)-dense universal sequence for \(F\) if \((a_n)^\infty_1\) contains an unbounded subsequence \((c_n)^\infty_1\) with \(\limsup_{n\to\infty}c_{n+1}/c_n=1\).
scientific article; zbMATH DE number 878230