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Can anybody help proving $ \mathbb{E}[X]\sqrt[3]{\mathbb{E}[X^3]}\leq\mathbb{E}[X^2] $ where $X$ is a nonnegative random variable with $\mathbb{E}[X^n]<\infty$ for finite $n$ (this differs from this post).

If possible, a counterexample would also be nice.

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This does not differ from your previous post, simply you did not get @mike's argument the first time. Briefly put:

The inequality to be proved cannot hold for every bounded nonnegative random variable.

To prove this, assume the inequality holds for every bounded nonnegative random variable and consider any of the unbounded counterexamples $X$ which were already given to you. Then, for every $k$, $X_k=\min\{X,k\}$ is bounded (in particular every moment of every $X_k$ is finite) hence, by your hypothesis, $ \mathbb E(X_k)\,\sqrt[3]{\mathbb E(X_k^3)}\leqslant\mathbb E(X_k^2). $ Now, let $k\to+\infty$. By the monotone convergence theorem, since $X_k\to X$ monotonically, one knows that $\mathbb E(X_k)\to\mathbb E(X)$, $\mathbb E(X_k^3)\to\mathbb E(X^3)$ and $\mathbb E(X_k^2)\to\mathbb E(X^2)$. These three limits are finite, hence you proved the bona fide inequality $ \mathbb E(X)\,\sqrt[3]{\mathbb E(X^3)}\leqslant\mathbb E(X^2), $ which you know to be false! Hence, your hypothesis was wrong, which is that the inequality to be proved holds for every bounded nonnegative random variable.

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    In case you are still interested: The tail of $X$ is not irrelevant for my purposes. Namely, I am particularly interested if the inequality is valid if the variable $X$ in the original inequality is (1) an upper truncated Gamma distribution (btw, not censored), or (2) has a density function of the type $x^k\exp(-ax-b\sqrt{x})$, (a>1, $b\geq0$, k>1). I just tried to pose the problem as general as possible. My apologies if I was unclear.2012-11-01