For what values of $0 < p,q < \infty$ is the following inequality of integrals valid?

Question

Let $m$ be the Lebesgue measure over $\mathbb{R}$ and let $f$ and $g$ be two nonnegative measurable functions defined on $[0,1]$ such that $f(x)g(x)\geq 1 \quad \forall x \in [0,1]$. It is not difficult to see that $ \left( \displaystyle\int_{[0,1]} f dm \right) \left( \displaystyle\int_{[0,1]} g dm \right) \geq 1$ by using Hölder's inequality for $p=q=2$ and the functions $\displaystyle\sqrt{f}$ and $\displaystyle\sqrt{g}$ and observing that $\sqrt{f(x)g(x)} \geq 1 \quad \forall x \in [0,1]$. But for what values of $0<p,q< \infty$ is it true that $\left( \displaystyle\int_{[0,1]}f^p \right)^{1/p} \left( \displaystyle\int_{[0,1]}g^q \right)^{1/q} \geq 1$? (supposing that both integrals are finite of course). It is clear from Hölder's inequality and from the fact that $\displaystyle\int_{[0,1]}fgdm \geq 1$ that the above inequality is true if $p,q>1$ and $\cfrac{1}{p}+ \cfrac{1}{q}=1$ and, because of the observation made above, the inequality is also valid if $p=q=1$, but what about all the other possible values for $p,q \in (0, \infty)$? A proof or a counterxample for the reamining cases or any hints or ideas for them would be greatly appreciated. Thanks in advance.

Answer 1

If $p,q$ are positive, then $1/p+1/q$ is positive, say equal to $a$. Since $$1\leqslant \int_{[0,1]} f(x)^{1/a}g(x)^{1/a}\mathrm dm ,$$ we can apply Hölder's inequality to the conjugate exponents $pa$ and $qa$.