Let \({\mathcal C}\) be the class of pdf's f on (0,\(\infty)\) such that, for each \(u>0\), f(uv)f(u/v) is completely monotone as a function of \(w=v+v^{-1}\). This class includes many familiar pdf's and is closed with respect to multiplication and division of independent rv's. Further, \({\mathcal C}\subset {\mathcal I}\), where \({\mathcal I}\) is the class of generalized Gamma convolutions (GGC) introduced by \textit{O. Thorin} [see Scand. Actuarial J. 1978, 141-149 (1978; Zbl 0392.60015)]. Moreover, \({\mathcal C}\) coincides with the class of pdf's of the form \[ C\cdot x^{\beta - 1}\cdot \prod^{N}_{i=1}(1+c_ ix)^{-\gamma_ i} \] (all parameters positive) or limits thereof. The Laplace transform \(\phi\) of a GGC is characterized by complete monotonicity of \(\phi\) (uv)\(\phi\) (u/v) as a function of w. This characterization has many consequences and applications. It follows that also the class \({\mathcal I}\) has simple multiplicative properties.