Thank you in advance for any input! The question is as follows.
$M_1(t)$ and $M_2(t)$ are the moment generating functions of two distributions indexed by (multi-dimensional) $\theta_1$ and $\theta_2$ in the same parametric family. Let's say the 1st moment are given by $\mu_1$ and $\mu_2$ (which are determined by $\theta_i$)
I'm trying to prove that if
$(1-t \mu_1) M_1(t) = (1-t \mu_2) M_2(t)$
then $M_1(t)$ and $M_2(t)$ must be exponential with MGF $(1-t \mu_i)$ where $\mu_i$ is the mean (i.e. 1/rate).
This is clear if $(1-t \mu_1) M_1(t) = (1-t \mu_2) M_2(t) = Constant = 1$
However, I'm worried that there exists some function $h(t,\theta_1,\theta_2) \neq 1$ such that $$(1-t \mu_1) M_1(t) = (1-t \mu_2) M_2(t)=h(t,\theta_1,\theta_2) \neq 1$$
From the first two moments, we can quickly conclude that $h(0,\theta_1,\theta_2) =1,$ $\frac{d}{dt} h(t,\theta_1,\theta_2)|_{t=0} =0$
Any suggestions on how to rule out this possibility that $h(t,\theta_1,\theta_2)\neq 1$ ? Would using a characteristic function help?
I've also attempted to prove that $\frac{(1-t \mu_2)}{(1-t \mu_1)} M_2(t)$ is not an MGF, but no success.
Again, greatly appreciate any tips or input!