Is there any necessary and sufficient conditions for a function to be the moment generating function of some random variables? In particularly, I'm working on this problem:
If we know $M(t)$ is a m.g.f of some r.v., then can $\dfrac{M(t)}{1+t}$ be a possible m.g.f for some other r.v.?
I know some similar problems of this kind have been discussed here before but I still don't know how to figure this one out. It satisfies $M(0)=1$ and $\dfrac{1}{1+t}$ doesn't look like a m.g.f of some common distribution(if it is then that's easy, since it can be viewed as sum of two independent r.v.). No way that I know works so far, so idk what can I do now.
Any help and hint is welcome and appreciated.