Ultra-log-concavity and the Choquet order

In the first part of the article we show that for any convex function $\phi$ the expression $\mathbb{E} \phi(X)$ is maximized, among all ultra-log-concave random variables supported in $\{0,\ldots, n\}$ with fixed mean $m$, for a binomial random variable $Y=\textbf{Bin}(n,m/n)$. As a consequence we prove the inequality $H(X)\leq H(Y)$. In the second part we develop the notion of discrete degrees of freedom of a log-concave sequence and use it to prove that the quantity $\mathbb{P}(X=\mathbb{E} X)$ is maximized, under fixed mean, for a Poisson distribution.