Prove that $E(X+Y)^p \le 2^p (E(X^p)+E(Y^p))$

Question

Suppose $X, Y$ are nonnegative random variables and $p \ge 0$. How can I prove that

$E(X+Y)^p \le 2^p (E(X^p)+E(Y^p)).$

I would appreciate any help.

Answer 1

Hint: try using the two inequalities $X+Y\leq 2\max\{X,Y\}$ and $\max\{X,Y\}\leq X+Y$.