# Littlewood's inequality for $L^p$ spaces

I have tried to prove the following inequality, but I couldn't do yet.

Prove the following interpolation estimate:

$$\| u\|_q \leq \| u\|_p^{\theta} \| u\|_r^{1- \theta}$$ where $$p≤q≤r$$, $$θ∈[0,1]$$ and $$\frac{1}{p} = \frac{\theta}{p} + \frac{1-\theta}{r}$$.

Note that $$\| u\|_q$$ denotes $$L^q$$ norm.

Analysis kills me. Any help will be appreciated.

Use Holder's inequality $$\|fg\|_a\le\|f\|_b\|g\|_c$$ where $$\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$$, on $$|u|=|u|^\theta|u|^{1-\theta}$$, with $$a=q$$, $$b=p/\theta$$, $$c=r/(1-\theta)$$, $$\|u\|_q\le\||u|^\theta\|_b\||u|^{1-\theta}\|_c=\|u\|_p^\theta\|u\|_r^{1-\theta}$$