Proving inequalities on $L^P$ spaces

Question

Suppose $p,q,r \in[1,\infty)$ and $ 1/r = 1/p +1/q$. Prove that

$$\|fg\|_r \leq \|f\|_p*\|g\|_q.$$

I am assuming that this proves involves using Hölder inequality , but so far I am unable to proceed in the proof. Maybe so because this is my first problem about using the Hölder/Minkowski inequaltiy.

Answer 1

Clearly $p,q>r$, so one may apply Holder's inequality to $|f|^r$, $|g|^r$ with $p'=\frac{p}r$, $q'=\frac{q}r$.