If $f \in L^{p}\left(\mathbb{R}^{n}\right) \cap L^{q}\left(\mathbb{R}^{n}\right),$ where $p<q,$ prove that $\|f\|_{r} \leq\left(\|f\|_{p}\right)^{\frac{1 / r-1 / q}{1 / p-1 / q}}\left(\|f\|_{q}\right)^{\frac{1 / p-1 / r}{1 / p-1 / q}}$ for any $p<r<q$. (I have already shown $f \in L^{r}\left(\mathbb{R}^{n}\right)$)
My approach: We have to show that $(\int |f|^r)^{\left(\frac{1}{r}\left(\frac{1}{p}-\frac{1}{q}\right)\right)}\leq(\int |f|^p)^{\left(\frac{1}{p}\left(\frac{1}{r}-\frac{1}{q}\right)\right)} (\int |f|^q)^{\left(\frac{1}{q}\left(\frac{1}{p}-\frac{1}{r}\right)\right)}$. But I'm unable to understand how to pick quantities the reciprocals of which sum to one, which would enable me to use Holder.
Write $|f|^{r}$ as $|f|^{a} |f|^{r-a}$ where $a= \frac {pq (q-r)} {q-p}$ and apply Holder's inequality with exponents $\frac p a$ and $ \frac q {r-a}$.