Question 71
Let $p$ be a positive number and $f : [0, 1] \to \mathbb{R}$ be a continuous concave function, such that $f(0) = 1$. Show that
$$
(p+1) \int_{0}^{1} x^{2p} f(x) \, dx + \frac{2p-1}{8p+4} \le \left( \int_{0}^{1} f(x) \, dx \right)^{2}
$$
and find when the equality is possible.
$$
(p+1) \int_{0}^{1} x^{2p} f(x) \, dx + \frac{2p-1}{8p+4} \le \left( \int_{0}^{1} f(x) \, dx \right)^{2}
$$
and find when the equality is possible.