$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ then Holder's inequality

Question

If $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ and $ f\in L_p $ $g\in L_q $ and $h\in L_r $ so how can I prove $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$

Answer 1

As one might expect, this works by applying Hölder's inequality twice:

Let $s=p$, $t=\frac{qr}{q+r}$. Then $\frac 1 s+\frac 1 t=1$, and Hölder's inequality yields $$ \|fgh\|_1\leq \|f\|_s\|gh\|_t=\|f\|_p\| |gh|^t \|_1^{\frac 1 t}. $$ Next let $\sigma=\frac q t=\frac{q+r}{r}$ and $\tau=\frac r t=\frac{q+r}{q}$. Then $\frac 1\sigma+\frac 1\tau=1$, and a second application of Hölder's inequality gives $$ \| |gh|^t \|_1^{\frac 1 t}\leq \| |g|^{t}\|_\sigma^{\frac 1 t}\| |h|^{t}\|_\tau^{\frac 1 t}=\| |g|^{t\sigma}\|_1^{\frac 1{t\sigma}}\| |h|^{t\tau}\|_1^{\frac 1{t\tau}}=\|g\|_q \|h\|_r. $$ Combine these two results to get the desired inequality.

To prove the generalized inequality for $n$ factors, just proceed by induction.