When I was solving an olympiad problem, I came out of this hypothetical inequality:
For bijective $f:[0,1]\rightarrow [0,1]$, continuous in $[0,1], f(0)=1, f(1)=0$, then we have: $$\int_0^1 xf(x)dx\int _0^1xf^{-1}(x)dx\ge \dfrac{2}{9}\left(\int_0^1 f(x)dx\right)^3$$
Apparently, the equality case is $f(x)=1-x$, or more generally, $f''=0$ (if the continuous condition is not necessary). In fact, there is a more general way to present the inequality. Note that for the region under $y=f(x)$ in $[0,1]$, the $x$-coordinate of centroid of region is determined by $\frac{\int_0^1 xf(x)dx}{\int_0^1 f(x)dx}$, similar for $y$-coordinate. So by this we could rephrase the statement into:
Given a bounded region $A$ in the first quadrant and the coordinates of its centroid are $(C_x,C_y)$, then $C_xC_y\ge \dfrac{2}{9}Area(A)$.
I am absolutely astonished by this result, but I have no idea how to verify it (maybe functional calculus?) It would be great if anyone could give some idea on this problem, even counterexamples!