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$X$ is a positive continuous random variable. $E[X^p]$ is the $p$-th moment of $X$, $p\ge2$. Is the following moment inequality valid? $E[X^p]\le (p-1)^{p/2}(E[X^2])^{p/2}$

If so, What is the name of this inequality, and how to prove it?

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    Try $p=1$. (extra characters)2012-03-20
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    Have you tried playing around with $E(X^p)=\int_0^\infty py^{p-1}P(X>y)dy$, which holds for $p>0$? As Byron mentions, it's not true for $p=1$, but maybe try looking at $1 and that $P(X>y)dy/E(X)$ is a probability measure along with Jenson's Inequality.2012-03-20
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    Let me up the ante:) For $1 consider the degenerate random variable $X\equiv 1$.2012-03-20
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    Sam, could you please explain your answer in more detail? I just modify p>=2 and p is integer. I really appreciate your answers!2012-03-24

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