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 $\frac{1}{q} = \frac{\theta}{p} + \frac{1-\theta}{r}$.
Note that $\| u\|_q $ denotes $L^q$ norm.
Analysis kills me. Any help will be appreciated.