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.