Is this inequality correct?

I've encountered this nice-looking inequality:

$$\left(A+B\right)^{p} \le p\left(A^{p}+B^{p}\right)$$

(p can irrational as well)

but I can't find a way to prove or disprove its correctness. I've tried using the binomial theorem, but it didn't seem it would lead me to the finish.

Could someone please tell me how to prove that?

Thank you very much!