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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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    Thanks for the clarification! The inequality is valid if $X$ is a Gamma distribution or the absolute value of the Normal distribution. Is the inequality true if $X$ is any thin tailed (no power law decay) distribution possibly truncated above and/or below?2012-11-01
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    The inequality is false even when restricting it to **bounded** distributions. Every bounded distribution is thin-tailed. Ergo.2012-11-01
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    Let me clarify a bit. Though the $X_k$ are thin-tailed in your monotone convergence theorem example, they are not truncated thin-tailed distributions. So I wonder if the inequality is true if $X$ is a truncated thin-tailed distribution.2012-11-01
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    You are not listening. Read carefully my answer, it **proves** that the answer to the question you keep asking is **NO**. Not *maybe*, not *yes in some cases*, just **NO**.2012-11-01
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    I am sorry but the $X_k$ are constructed with a $fat$-tailed $X$. Not every distribution can be constructed in this way. So I do not understand why your answer implies that the inequality is also violated if $X$ is $thin$-tailed, and $X_k$ is still constructed by $\min\{X,k\}$.2012-11-01
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    The inequality fails for some $X_k$, say $X_{223}$, and **this random variable** $X_{223}$ has the thinnest possible tail. The fact that $X_{223}$ is defined using $X$ and that $X$ is fat-tailed is irrelevant (by the way $X_{223}$ is also $\min\{Y,223\}$ or $\min\{Y,456\}$ or $Y$, for a host of random variables $Y$ with thin tails).2012-11-01
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/6295/discussion-between-alexander-and-did)2012-11-01
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    Let us not. $ $2012-11-01
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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