I have to prove that : Let $p \in [0,1], A,B > 0$ and $\alpha \in [0,1]$. Then :
$$(\alpha A+(1−\alpha)B)^p \ge \alpha A^p +(1−\alpha )B^p$$
I have the hint that : $X \ge Y \iff X−Y \ge 0$.
I also know that the solution of this problem reside in the fact of reformulating the problem in 1 or several inequalities, and that I can't use convexity and concavity in the matrix space or I need to define it