Suppose $f$ is a continuous function on $[1,\infty)$ such that$$\int_1^{\infty} f(x)^2 dx <\infty$$
Is it true that$$\int_{1}^{\infty} \frac{1}{r^2} \left( \int_{1}^r f(x) dx\right)^2 dr < \infty$$
Any ideas on how to prove it or come up with a counterexample?
Note that if the domain was finite, then the above is true simply by the Cauchy-Schwartz (or by Jensen's inequality).