You know that , this is the moment generating function. Or maybe you know , which is the same with . Anyway, you can compute this, even if you don't know it already.
On the other hand, since (you should have mentioned it) X and Y are independent, the conditional distribution of Y given X=x is the same as the distribution of Y. Therefore, for all , . Then , and the last expectation can be computed using the moment generating function again.
The last inequalities can also be written .