Let $f:\mathbb R \to \mathbb R$ be a $C^1$ function and suppose that $0<f'(t)\leq 1$ on $[0,1]$ and $f(0)=0$. Does the inequality $$\left(\int_0^1 f(t)dt\right)^2 \leq \int_0^1f^3(t)dt$$ hold? By $f^3(t)$ I mean $(f\circ f\circ f)(t)$.
This question appeared on a test I took but there was no information about what $f^3(t)$ means. If I suppose that $f^3(t)=(f(t))^3$ then the inequality is true and very simple to prove, but as far as I know $f^3$ is the composition of $f$ three times.