Let $f:[0,1]\to [0,1]$ be a non-decreasing concave function, such that $f(0)=0,f(1)=1$. Prove or disprove that :
$$ \int\_{0}^{1}(f(x)f^{-1}(x))^2\,\mathrm{d}x\ge \frac{1}{12}$$
A friend posed this to me. He hopes to have solved it, but he is not sure. Can someone help? Thanks.