A function $f(x)$ is completely monotonic if $(-1)^nf^{(n)}(x) \geq 0$ for all $n \geq 0$ and all $x > 0$. These functions are characterized by Bernstein's theorem and are well studied in Widder's book on Laplace transforms where
$g(t) = \int_0^\infty e^{-tp} d\alpha(p)$ where $\alpha(p)$ is bounded and non-decreasing. Upon application of the Laplace transform, Widder states we may write $G(s)$ as
$G(s) = \int_0^\infty \frac{d\alpha(t)}{s+t}$
My question is, suppose I had a function $H(s)$ such that
$H(s) = sG(s) = \int_0^\infty s\frac{d\alpha(t)}{s+t}$.
Does anyone know if such functions also have a theory similar to that of the completely monotonic functions? I was having difficulty finding resources via Google Scholar on this class of functions.