I am reading the book of Püss (Evolutionary Integral Equations and Applications pg. 101) about subordination principle and it have one thing that i dont understand. If $c$ is a completly positive function how we can conclude that the function
$$h(\tau; \lambda)=\frac{e^{-\frac{\tau}{\widehat{c}(\lambda)}}}{\lambda \widehat{c}(\lambda)}$$
is completly monotonic in $(0,\infty)$ for all $\tau >0$? Here $\widehat{c}$ is the Laplace Transform of $c$. I am trying as follow... Since $c$ is completly positive function, exist a function $b:[0,\infty)\to \mathbb{R}_+$ suth that $c=b'$ (derivative of $b$) such that $db$ is positive measure, that is, exist a creep function $k:(0,\infty)\to \mathbb{R}_+$ such that $$\widehat{db}(\lambda)\widehat{dk}(\lambda)=\frac{1}{\lambda}$$ By the proposition 4.5 of Püss the function $e^{-\frac{\tau}{\widehat{c}(\lambda)}}$ is completly monotonic. We obtain too that $\frac{1}{\widehat{c}(\lambda)}$ is a Bernstein function, then $\widehat{c}(\lambda)$ is completly monotonic. But I cant conclude that $h$ is completly monotonic.
You forgot say that $c$ is completly continuous. But lets go, note that $\widehat{db}(\lambda)=\widehat{c}(\lambda), \ \lambda>0$ so $$\widehat{dk}(\lambda)=\frac{1}{\lambda \widehat{c}(\lambda)}, \ \lambda>0.$$ Since $k$ is a creep function, by the Bernstein Theorem $$\lambda \mapsto \frac{1}{\lambda \widehat{c}(\lambda)}$$ is completly monotonic. But the completly monotonic functions are close to multiplication, hence $h(\tau;\lambda)$ is completly monotonic on $(0,\infty)$ for all $\tau>0$.