Assume $f$ is $C^1([0,1])$ and $f(0)=0$ and $ 0 < f' \le 1 $ then I want to show that $$\int_{0}^{1} f^3 dx < \left( \int_{0}^{1} f dx \right)^2 $$
my tries : I want to use a similar way like the proof of the poincare inequality
$f(x)^3=\int_0^1 3 f'(x) f^2 (x) dx $ now I use the bound of $f'$ and $f$ and Holder inequality and integrating from both side of the inequalities but i reach inequalities different of the asked one.
Can some one help me. thanks.