Cauchy-Schwarz-like inequality of integrals

Question

Let $f,g,$ be integrable on $[a,b]$. Prove that $$\int_a^b(fg)^2\le\int_a^bf^2\int_a^bg^2$$

I know that from Cauchy-Schwarz we have $$\left(\int_a^bfg\right)^2\le\int_a^bf^2\int_a^bg^2$$

so if we showed that $$\int_a^b(fg)^2\le\left(\int_a^bfg\right)^2$$ we would be done. But I don't think this is even true in general, so this method doesn't seem to lead anywhere. Is there another approach to this problem that I'm missing?

Edit: The inequality seems to be false. Perhaps the inequality was given incorrectly.

Answer 1

This doesn't seem to be true in general. For example:

Take $f(x)=1$ for $x \in [0,\frac{1}{2}]$

$g(x)=x$ (same interval.)

Then, your LHS= $\frac{1}{24}$, RHS is $\frac{1}{48}$