TITLE: An inequality involving $\mathbb{E}[|X|],\,\mathbb{E}[X^2],\,\mathbb{E}[X^4]$
QUESTION [1 upvotes]: My question is an exercise which appears in a book on probability theory. Thank for helping.

Let $X$ be a random variable with $\mathbb{E}(X^2)=1$,
  $0<\mathbb{E}(X^4)<+\infty$. Prove that: $$\mathbb{E}[\left| X \right|]
 \geqslant \frac{1}{{\sqrt {\mathbb{E}[ X^4 ]} }}.$$

REPLY [1 votes]: $$\mathbb{E}[X^4]^{1/2}\mathbb{E}[|X|]=\mathbb{E}[X^4]^{1/4}\mathbb{E}[X^4]^{1/4}\mathbb{E}[|X|]$$
So, by the generalized Holder's inequality ($r=\frac{2}{3}$ since $\frac{1}{4}+\frac{1}{4}+1=\frac{3}{2}$) we have:
$$ \mathbb{E}[X^4]^{1/2}\mathbb{E}[|X|]\geq \mathbb{E}[X^2]^{\frac{3}{2}}=1.$$
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