I am reading a book where it is written:
For any $n \geq 1$, $E(|x|^n) \geq (E(|x|))^n$
I understand that it is simple apllication of Jensen' inequality for function $f(x) = |x|^n$, but I have some question about right side of the equation. Shouldn't it be $(|E(x)|)^n$ instead of $(E(|x|))^n$? Apparently these are quite different.
I believe that it is an application of Jensen's inequality to the convex function $\varphi(t)=t^n$ and the integrand $x \mapsto |x|$: $$ \left( \int |x| \, d\mu(x) \right)^n \leq \int |x|^n \, d\mu(x). $$