The random variable X is stochastically dominated by Y. Therefore, for k = 1,2,...
E[X^k] > E[Y^k]
Show
| \left(\frac{E[X^(k+1)]/E[X^k]) }{ (E[Y^(k+1)]/E[Y^k]) } \right| < 1
If needed, one can impose positivity of Y, continuity of X and Y, and/or finiteness of moments of X and Y.

Thanks in advance,
Sander