Let $X$ be a continuous random variable with support $(0,a)$ and cumulative distribution function $F$ and a density function $f$ that is positive on the support. ($a$ can be taken to be finite or as infinity.)
I am interested in finding an answer to the following question: When does the following inequality hold?
$$ \int\limits_{0}^{a}\left( 2F^{4}\left( x\right) -3F^{3}\left( x\right) +F^{2}\left( x\right) \right) dx\leq 0 $$
For example, when $X$ is uniformly distributed, this holds. (I have several other examples that satisfy this. Although I do not have an explicit example in which this does not hold, I do not think it always holds, of course.)
I am interested in generalizing the class of distributions that satisfy this relation. Something like "when $F$ is concave" (I actually conjecture that this is true but I do not have a proof.) or "when density has certain property" and so on.