Slow increase of Stieltjes integrals (Q2768774)

This paper deals with sufficient conditions for slow increase of Stieltjes integrals. In particular, the author proves the following result. Let \(l\) be a positive continuous slowly increasing function on \([a,+\infty)\), let \(l_1\) and \(l_2\) be positive continuous functions on \([a,+\infty)\) such that \(l_1(x)\leq l_2(x)\) for all \(x\geq a\) and let NEWLINE\[NEWLINE{l_2(l^{-1}(2l(2x)))\over l_1(x)}\ln{l(2x)\over l(x)}\to 0,\quad x\to+\infty.NEWLINE\]NEWLINE Let \(L(x)=\int_{a}^{x}\psi(t) dl(t)\), where \(\psi\) is a positive nondecreasing function on \([a,+\infty)\). If \(l_1(x)\leq L(x)\leq l_2(x),\;x\geq a\), then the function \(L\) is slowly increasing on \([a,+\infty)\).