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.

From Jensen, we can obtain,
$$a^{\theta}b^{1 - \theta} \leq \theta a + (1 - \theta) b$$
for $a, b \geq 0$ and $0 \leq \theta \leq 1$.
Substitute $a = (1 - \theta)a$ and $b = \theta b$, and we get,
$$((1 - \theta)a)^{\theta}(\theta b)^{1 - \theta} \leq \theta (1 - \theta) a + (1 - \theta) \theta b$$
which gives,
$$a^{\theta}b^{1 - \theta} \leq \theta^\theta (1 - \theta)^{1 - \theta} (a + b)$$
The rest should follow.
