SCHWEITZER 1973 Problem 6

If $f$ is a nonnegative, continuous, concave function on the closed interval $[0,1]$ such that $f(0)=1$ , then $$\int_0^1 xf(x)dx \leq \frac 23 \left[ \int_0^1 f(x)dx \right]^2.$$

Z. Daroczy