  We consider the following question: if a function of the form
$\int_0^{\infty}\varphi(t)\, e^{-xt}dt$ is completely monotonic, is it then
$\varphi\ge0$? It turns out that the question is related to a moment problem.
In the end we apply those results to answer some questions concerning complete
monotonicity of certain functions raised in F. Qi and R. Agarwal, \textit{On
complete monotonicity for several classes of functions related to ratios of
gamma functions}, J. Inequal. Appl. (2019).
