$(E,A,\mu)$ is a finite measure space, $f$ and $g$ are both positive and measurable functions from $E$ to $\mathbb{R}$ such that $fg \geq 1$
by holder we have $$(\int_E f^2 \ d\mu)(\int_E g^2\ d\mu) \geq (\int_E fg \ d\mu)^2 \geq (\int_E \ d\mu)^2 \geq \mu^2(E)$$
how about this one though : $\int_E f\ d\mu\int_E g\ d\mu \geq \mu^2(E)$
if $f = h^2, \, g = w^2$ then $|h||w| \geq 1$ and $|h|,\,|w| \geq 0$ and measurable therefore by what precedes
$\int_E f\ d\mu\int_E g\ d\mu = (\int_E h^2 \ d\mu)(\int_E w^2\ d\mu) \geq \mu^2(E)$
Is my approach correct?