Suppose that $f$ and $g$ are strictly increasing and continuous functions from $[0,1]$ to $[0,1]$, they are bijective. I'm trying to find a condition which makes the following inequality holds: $$\big(f(x)-g(x)\big)^2\leq\int^1_0\big(f(x)-g(y)\big)^2dy,~\forall x\in[0,1].$$ If the $f(x)$ inside the integral part were $f(y)$, the inequality should hold for any $f$ and $g$ by Jensen's inequality, but unfortunately, I have $f(x)$ there..
Fixing $g(x)=x$, here are some cases. If $f(x)=x^2$, the inequality is true for all $x\in[0,1]$. However, if $f(x)=x^3$, there are some $x$'s which break the inequality.
Is there any convexity or concavity approach to find out what kind of condition on $f$ and $g$ make the inequality be true?