TITLE: Why $L^{r}(X)\cap L^{t}(X)\subset L^{s}(X)$ for $1<r<s<t$?
QUESTION [5 upvotes]: I am working on this homework problem, and I am totally stuck:
Let $(X,\mu)$ be a measure space, and let $1 \leq r < s < t < \infty$.  Prove that there exist constants $\alpha,\beta>0$ so that
$$
\|f\|_s \;\leq\; \|f\|_r^\alpha\,\|f\|_t^\beta
$$
for any measurable function $f\colon X\to\mathbb{R}$.  Use this to show that
$$
L^r(X) \cap L^t(X) \,\subset\, L^s(X).
$$
I know this is supposed not to be difficult. But I cannot solve it.

REPLY [5 votes]: Hint: Write $\frac{1}{s} = \frac{\alpha}{r} + \frac{\beta}{t}$ and apply Hölder's inequality.