How to prove such an elementary inequality The inequality is the following:
$a^ \theta b^ {1-\theta}$ $\leq$ $[\theta ^ \theta (1-\theta)^ {1-\theta}]^{1/p}(a^p+b^p)^{1/p}$, where $\theta \in [0,1]$, $a,b$ are nonnegative.
This inequality is used to give a sharper constant in the proof of an embedding theorem in Sobolev spaces. Here is the link https://www.math.ucdavis.edu/~hunter/pdes/ch3.pdf. On page 66, the author used the inequality to give a sharper estimate, but he didn't give a proof of this inequality. Actually, 
$a^ \theta b^ {1-\theta}$ $\leq$ $(a^p+b^p)^{1/p}$ is obvious (this is my first try), and enough to prove the embedding theorem, but it is always interesting to give a sharper inequality. 
I tried to prove this seemingly elementary inequality, but I'm really not good at it. Can anyone give a smart answer? Any hint would be appreciated.

By the AM/GM inequality,
$$ \left(\frac a\theta\right)^\theta \left(\frac b{1-\theta}\right)^{1-\theta}
\le \theta\left(\frac a\theta\right) + (1-\theta)\left(\frac b{1-\theta}\right)
= a+b
$$
Now replace $a$ and $b$ with $a^p$ and $b^p$.
