Hölder' inequality for random variables

Question

$X_n$ and $X$ is integrable random variable, for some $p>1$, we have $|X_n-X|^p\leq 2^{p-1}(|X_n|^p+|X|^p)$.

Does anyone know how to derive this inequality? The solution tells me it's Hölder's inequality, but I cannot see it. Thanks a lot!

Answer 1

One way to see this holds is that the map $x \mapsto |x|^p$ is convex (the second derivative is positive) for $p \geq 1$. Thus for all $x,y \in \mathbb{R}$

$$\frac{1}{2^p}|x+y|^p=\left|\frac{x}{2} + \frac{y}{2}\right|^p \leq \frac{1}{2}|x|^p+ \frac{1}{2}|y|^p$$ and thus

$$|x+y|^p \leq 2^{p-1}(|x|^p + |y|^p)$$

from which your inequality follows (subsitute $-y$ instead for $y$ in this formula).