Let $p \ge 1$ be an integer. Does there exist a constant $C\_p$ such that for every random variable $X \ge 0$,
$$
\mathbb{E} \left[ \left(X - \mathbb{E} \left[ X \right] \right)^{2p} \right] \le C\_p \mathbb{E} \left[ \left(X^p - \mathbb{E} \left[ X^p \right] \right)^{2} \right] \ \ ?
$$



Is this simple-looking moment inequality true?
The answer is yes. Let $Y$ be an independent random variable distributed as $X$. We have
$$
\|X - \mathbb{E}[X]\|\_{L^{2p}} = \|\mathbb{E}\left[X - Y \, | \, X\right]\|\_{L^{2p}} \le \|X - Y\|\_{L^{2p}}.
$$
Moreover, there exists a constant $C\_p$ such that for every $x,y \ge 0$,
$$
|x-y|^p \le C\_p |x^p - y^p|.
$$
Indeed, it suffices to verify this for $x = 1$ and $y \in [0,1]$ by homogeneity and symmetry. This is then a simple exercise. As a consequence, we deduce
$$
\|X - \mathbb{E}[X]\|\_{L^{2p}} \le C\_p \||X^p - Y^p|^{1/p}\|\_{L^{2p}} = C\_p \|X^p - Y^p\|\_{L^{2}}^{1/p}.
$$
We conclude by the triangle inequality.


