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

(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.