How to show that if $\mathbb{E}(X^4) < \infty$, then $\mathbb{E}(X^2) < \infty$?

Question

Let $\mu = \mathbb{E}X$ and $\sigma ^2 = Var(X)$ and $X_1, ..., X_n$ a sample. How do I show that if $\mathbb{E}(X^4) < \infty$, then $\mathbb{E}(X^2) < \infty$ using Jensen's inequality?

Answer 1

This is pretty much a direct application of Jensen's inequality. Let $g(x)=x^2$ and $\varphi(x)=x^2$. Then $\varphi$ is convex so by Jensen's inequality, $$ \varphi \left(\int_\Omega g \mathrm{d}\mu\right)\leq \int_\Omega \varphi \circ g\,\mathrm{d}\mu \ $$ which shows $[E(X^2)]^2\leq E[X^4]<\infty.$