Let $(X;A;\mu)$ be a measure space, choose real numbers $1 \leq r < p < s < \infty,$ and let $0 < \lambda < 1$ such that $$\frac \lambda r+ \frac {1-\lambda} {s}=\frac1p $$ Prove that every measurable function $f : X \rightarrow \mathbb{R}$ satisfies the inequality: $$\Vert f\Vert_p\leq \Vert f\Vert_r^\lambda \Vert f \Vert_s^{1-\lambda}. $$
I assume I have to use Hölder's inequality, but I don't see exactly how. Any help would be great.