Exercise 8, § 2

Let $f$ be a regulated function and $\geqslant 0$ on an interval ]$a, +\infty$*, and let $g$ be a differentiable increasing function defined on ]$a, +\infty$*, and such that $g(x) - x \geqslant \lambda > 0$ for all $x \geqslant a$.

Show that if one has $f(g(x)) g'(x) \leq k f(x)$ with $k < 1$ (resp. $f(g(x)) g'(x) \geq k f(x)$ with $k > 1$), then the integral $\int_a^{+\infty} f(t) \, dt$ converges (resp. is equal to $+\infty$) (Ermakoff’s criterion: denoting the $n^{th}$ iterate of $g$ by $g^n$, consider the integral $\int_0^{g^n(a)} f(t) \, dt$ and let $n$ increase indefinitely).