Let $\frac{1}{r}=\frac{\theta}{p}+\frac{1-\theta}{q}$ where $1\leq p<r<q\leq \infty$ and $\theta \in ]0,1[$, as well as $f \in L^{p} \cap L^{q}$. Show that $||f||_{r}\leq||f||_{p}^{\theta}\times||f||_{q}^{1-\theta}$
I get to the point:
$||f||_{r}=...=((\int_{\mathbb R}|f|^{r}d\mu)^{\frac{1}{p}})^{\theta}\times((\int_{\mathbb R}|f|^{r}d\mu)^{\frac{1}{q}})^{1-\theta}$
but this leads to a dead end. Any hints?
We have by the Holder ineq that $$\Vert f\Vert_r=\Big\Vert \vert f\vert ^{\theta}\vert f\vert ^{1-\theta}\Big\Vert_r\leq \Big\Vert \vert f\vert ^\theta\Big\Vert_{p/\theta} \Big\Vert \vert f\vert ^{1-\theta}\Big\Vert_{q/(1-\theta)}.$$