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If $f$ is in $L^2$, is its average in $L^2$? [duplicate]

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).


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