Question 154
Let $f : [0, 1] \to [0, \infty)$ be a convex function with $f(0) = 0$. Show that
$$
\left( \int_{0}^{1} f(x) \, dx \right)^{2} \leq \frac{3}{4} \int_{0}^{1} f^{2}(x) \, dx .
$$
$$
\left( \int_{0}^{1} f(x) \, dx \right)^{2} \leq \frac{3}{4} \int_{0}^{1} f^{2}(x) \, dx .
$$