The following problem might have something to do with the coefficients of the moment generating function. But I do not see how to prove it, nor do I have a counterexample.
Given that the positive random variable $X$ is first order stochastically dominated by $Y$ $(0\preceq X\preceq Y)$, we know $$ E[\exp(-X)]\geq E[\exp(-Y)] $$ In other words, $$ \sum_{i=0}^\infty \frac{E[(-X)^i]}{i!} \geq \sum_{i=0}^\infty \frac{E[(-Y)^i]}{i!} $$
Is the following true for any $k$ and $0\preceq X\preceq Y$ $$\sum_{i=0}^\infty \frac{E[(-X)^i]}{(k+i)!} \geq \sum_{i=0}^\infty \frac{E[(-Y)^i]}{(k+i)!}$$ One may assume that $k$ is even if needed.