TITLE: Hölder's inequality with three functions
QUESTION [30 upvotes]: Let $p,q,r \in (1,\infty)$ with $1/p+1/q+1/r=1$. Prove that for every functions
  $f \in L^p(\mathbb{R})$, $g \in L^q(\mathbb{R})$,and $h \in L^r(\mathbb{R})$ 
  $$\int_{\mathbb{R}} |fgh|\leq \|f\|_p\centerdot \|g\|_q \centerdot\|h\|_r.$$

I was going to use Hölder's inequality by letting  $1/p+1/q= 1/(pq/p+q)$ and WLOG 
let $p<q$ so that $L_q(\mathbb{R})\subseteq L_p(\mathbb{R})$, but I cannot use this inclusion because $\mathbb{R}$ does not have finite measure. 
Would you please help me if you have any other method to approach this problem?

REPLY [29 votes]: We can use a generalized AM-GM inequality to deduce that if $1/p+1/q+1/r=1$, then
$$abc\le\frac{a^p}{p}+\frac{b^q}{q}+\frac{c^r}{r}$$
for nonnegative $a,b,c$. Let $a=|f(x)|/\|f\|_p,\,b=|g(x)|/\|g\|_q,\,c=|h(x)|/\|h\|_r$, and then integrate both sides of the inequality over $\mathbb{R}$ to obtain
$$\frac{\|fgh\|_1}{\|f\|_p\|g\|_q\|h\|_r}\le\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1.$$
Multiply out and you have Holder's inequality for three functions.