I have a question about the relation between concavity/convexity and a sum of conditional expectations.
Especially, suppose $X$ has a CDF $F$ and pdf $f$ with support $[0,1]$. Will concavity/convexity of $F$ decides the direction of the below inequality? $$x+\mathbb E[X]\geq\mathbb E[X|X>x]+\mathbb E[X|X\leq x],~\forall x\in(0,1)$$ or $$x+\mathbb E[X]\leq\mathbb E[X|X>x]+\mathbb E[X|X\leq x],~\forall x\in(0,1).$$
Here, by convexity/concavity of $F$, I mean $f'(x)\geq0$ or $f'(x)\leq 0$ for all $x\in[0,1]$.
I have tried $F(x)=x^a,~a>0$
And found the below inequality holds for $a<1$, whereas the above inequality is true for $a>1$. and equality holds for $a=1$.
Are there any connection between convex/concave $F$ and the above relation?
or, equivalently, can we show the sign of $$x-\big(F(x)\mathbb E[X|X>x]+(1-F(x))\mathbb E[X|X<x]\big)$$ depends on convexity of $F$?
Any help will be really appreciated.