Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform, $$ f(s)=\int_0^\infty e^{-st}d\mu(t). $$ For example, $$ f(s)=e^{-e^{-s}}=\sum_{t=0}^\infty \frac{(-1)^t}{t!}e^{-st}\geq 0. $$
In my case, $d\mu$ is always supported at discrete points, but results for nicer $d\mu$ are also helpful.
Given this information, what necessary (and, ideally, sufficient) conditions can one derive on $d\mu$?
This can be compared to a similar problem, where one assumes $\mathrm{Re} \,f(s)\geq 0$ for $\mathrm{Re}\,s\geq 0$, which seem to result in the requirement that $d\mu(t)+d\mu(-t)$ is non-negative-definite distribution. (See e.g. this paper by Konig and Zemanian)
