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Let $X$ be a non negative random variable. Prove that $$\mathbb{E}[X] < (\mathbb{E}[X^2])^{1/2} < (\mathbb{E}[X^3])^{1/3} < \cdots$$ $\mathbb{E}[X]$ stands for expectation value of a random variable X

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  • $\begingroup$ This is false, consider the example $X=0$ with prob 1. You get the incorrect inequality $0<0<0<...$ $\endgroup$
    – Michael
    Jul 2 '20 at 1:37
  • $\begingroup$ This is a direct result of the Power Mean inequality. $\endgroup$ Jul 2 '20 at 1:56
  • $\begingroup$ math.stackexchange.com/q/1289607/321264 $\endgroup$ Jul 2 '20 at 7:44
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Hint: For $0 < p < q$ formulate this as $\left(\mathbb E[Y]\right)^{q/p} \le \mathbb E[Y^{q/p}]$ where $Y = X^p$, and use Jensen's inequality.

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