Poisson processes and a log-concave Bernstein theorem

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in terms of log-concavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the Pr\'ekopa-Leindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.