  We establish a new connection between the class of Nevanlinna-Pick functions
and the one of the exponents associated to spectrally negative L\'evy
processes. As a consequence, we compute the characteristics related to some
hyperbolic functions and we show a property of temporal complete monotonicity,
similar to the one obtained via the Lamperti transformation by Bertoin \& Yor
({\it On subordinators, self-similar Markov processes and some factorizations
of the exponential variable}, Elect. Comm. in Probab., vol. 6, pp. 95--106,
2001) for self-similar Markov processes. More precisely, we show the remarkable
fact that for a subordinator $\xi$, the function $t \mapsto t^n \,
\er[\xi_t^{-p}]$ is , depending on the values of the exponents $n=0,1,2,\;
p>-1$, or a Bernstein function or a completely monotone function. In
particular, $\xi$ is the inverse time subordinator of a spectrally negative
L\'evy process, if, and only if, for some $\,p\geq 1$, the function $t \mapsto
t \, \er[\xi_t^{-p}]$ is a Stieltjes transform. Finally, we clarify to which
extent Nevanlinna-Pick functions are related to free-probability and to
Voiculescu transforms, and we provide an inversion procedure.
