  We find the best possible constant $C$ in the inequality
$\|\varphi\|_{L^r}\leq
C\|\varphi\|_{L^p}^{\frac{p}{r}}\|\varphi\|_{\mathrm{BMO}}^{1-\frac{p}{r}}$,
where $2 \leq r$ and $p < r$. We employ the Bellman function technique to solve
this problem in the case of an interval and then transfer our results to the
circle and the line.
