Given some $p \in [0,1]$, and some nonnegative, integrable random variable $X$. I claim that
$$\mathbb E [X^{p}]\leq \mathbb E [X]^{p}.$$
My line of reasoning:
The map $f:[0,\infty)\to \mathbb R,\; f(x):=x^{p}$, is concave (still working on a proof for this; any ideas?).
If we have concavity of $f$, then we may proceed as follows
$$ \mathbb E[X^{p}]=\mathbb E[f(X)]\leq f(\mathbb E[X])=\mathbb E[X]^{p}.$$
Is my reasoning fine?