Generalized Hölder inequality
Solution 1:
Hint: $$ 1 = \frac{r}{p}+\frac{r}{q} = \frac{1}{\frac{p}{r}}+\frac{1}{\frac{q}{r}} $$
Solution 2:
Using the standard Hölder inequality: $$ \|fg\|_r^r=\|f^rg^r\|_1\le\|f^r\|_{p/r}\|g^r\|_{q/r}=\|f\|_p^r\|g\|_q^r $$ Since $\frac{r}{p}+\frac{r}{q}=1$.