# Is the expectation $E[\xi U'(\xi)]$ finite?

I encounter the following problem today. It seems a simple question.

Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:

(1) $U$ is concave, continuous, and strictly increasing,

(2) $\limsup_{x\rightarrow +\infty}\dfrac{xU'(x)}{U(x)} <1.$

(3) $U'(0+) = +\infty, \mbox{ and }U'(+\infty) = 0.$

Is the following statement true?

$\bf Claim:$ For any non-negative random variable $\xi,$ if $E[U(\xi)] < +\infty,$ then we have $$E[\xi U'(\xi)] < +\infty.$$

$\bf Remark:$ If $U(0) >-\infty,$ it is trivial if one notices that $$0\leq \xi U'(\xi) \leq U(\xi) - U(0).$$ But in general, the property $U(0) >-\infty$ does NOT hold. For example $\ln(x).$

Any comment and suggestion are welcome. Thanks.

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The claim is wrong: try the random variable $\xi$ with density at $x$ proportional to $g(x)=\mathrm e^{-1/x}$ when $x\to0^+$ and the function $U$ defined by $U(x)=-1/g(x)$ for every $x\gt0$ .