In the article "CONCENTRATION OF THE INFORMATION IN DATA WITH
LOG-CONCAVE DISTRIBUTIONS" of Bobkov and Madiman, it is written the 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 s given, but I don't understand how the resulat follows from the reference.
Also, it seems quite hard to prove, and the problem where thoose variables came from is said to be "easy" (it's the one dimension optimal matching problem), so I start to feel like I have issunderstood something.

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