I'm currently running into the following inequality in a measure-theory related proof that are related to moment-generating functions and their derivatives.
Let $X \in \mathbb{R}_+$ and $h \in \mathbb{R}$. Show that:
$\cfrac{e^{hX} - 1}{h} \le X e^{hX}$
I'm fairly more conditions are required on $h$ for the inequality to hold (such as $h\ne 0$, though none were stated).
Other facts that I have are that may be relevant are: The quantity on the right is most likely the derivative of $e^{hX}$.
- $X$ is actually a non-negative random variable.
- $\mathbb{E}(e^{rX}) < \infty$ for $r\in[-\infty,s]$ where $s>0$.
- $\mathbb{E}(X^k) < \infty$ for $k>0$.
- $\mathbb{E}(X^k e^{rX}) < \infty$ for $k>0$ and $r\in[-\infty,s]$.