Is there any characterization of log-concavity (of probability distributions) in terms of the moment-generating function or characteristic function?
Specifically, I want to prove some random variable (real-valued) has a log-concave distribution. However, finding the pdf is a mess; but the MGF $M_X$ of $X$ is rather nice. Is there some property of $M_X$ which would imply log-concavity of (the distribution of) $X$?