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There is a Young inequality that says: given $a,b \in \mathbb{R}$ and $p,q> 1$ such that $\frac{1}{p} + \frac{1}{q}=1$ we have:

$$|ab| \leq \frac{1}{p}|a|^p + \frac{1}{q} |b|^q$$

The question is: given two real functions $f,g: [a,b] \to \mathbb{R}$ (let's take them continuous to avoid integrability problems) the inequality above should hold for every $x$ and thus pass to the integral (at least I don't see why it shouldn't). So I should have something like:

$$\int_{a}^{b} |f(x)g(x)| dx \leq \frac{1}{p} ||f||_{L^p} ^p + \frac{1}{q} ||g||_{L^q}^q$$

But I have never seen this, in this case I often use Holder for instance. Is this inequality true?

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It is true. The reason why you see just Holder's used more often, is because your inequality is equivalent to applying Holder's and Young's in succession. Indeed: $$\int_a^b |f(x)g(x)|dx\leq \|f\|_{L^p}\|g\|_{L^q}\leq\frac{1}{p}\|f\|_{L^p}^p+\frac{1}{q}\|g\|_{L^q}^q $$ The first inequality is Holder's while the second is Young's with $a=\|f\|_{L^p}$ and $b=\|g\|_{L^q}$.

Most of the times you just need Holder's inequality so you stop at the first step.

Also, Holder's inequality works (and is very often used) for the case $p=1,q=\infty$ (or viceversa) while this one does not.

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