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Let $F$ be the CDF for a Beta distribution, $$F(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_{0}^{x}t^{a-1}(1-t)^{b-1}\,dt$$ with $a,b\geq 1$. Is it true that $$\frac{b}{a+b} \leq\int_0^1\sqrt{F(x)}...

I made the following claim, which I now see that I don't know how to prove. Can anyone prove it? Claim. Let $f$ be a concave and non-negative function on $[0,1]$ with $$\int_0^1 f = 1,$$ $$\int_0^1 |f-...