In the article "Concentration of the information in data with Log-concave distributions" of Bobkov and Madiman, it is written that if $X$ is a positive random variable following a log concave distribution of order $p$, then one has $V(X) \leq \frac{E(X)^2}{p}$. A reference is given, but I don't understand how the result follows from the reference. Also, it seems quite hard to prove, and the problem where those variables came from is said to be "easy" (it's the one dimension optimal matching problem), so I start to feel like I have misunderstood something.

Have you seen this inequality? Is it possible to give a relatively short proof?