Suppose $F$ is a cdf with support $[0,1]$. Let $0\leq a \leq b \leq 1$.
I am interested in distributions that satisfy the following inequality:
$$\mathbb{E}[x F(x) \mid x \in [a,b]] \geq \mathbb{E}[x F(x) \mid x \in [0,a]\cup [b,1] ] .$$
It seems to have something to do with concavity but I haven't been able to pin down the exact relationship.