Question 52
Let $f:[0,1]\longrightarrow[0,1]$ be a continuous bijection such that $f(0)=0$. Prove that, for every $\alpha\geq0$,
$$
(\alpha+2)\int_0^1 x^\alpha\bigl(f(x)+f^{-1}(x)\bigr)\,\mathrm{d}x\leq2.
$$
$$
(\alpha+2)\int_0^1 x^\alpha\bigl(f(x)+f^{-1}(x)\bigr)\,\mathrm{d}x\leq2.
$$