Let $X,Y$ be be random variables whose moment generating functions $s\mapsto \mathbb{E}(e^{sX})$ exist and agree on either the interval $(-\delta,0]$ or on the interval $[0,\delta)$ for some $\delta > 0$. Do $X$ and $Y$ have the same distribution?
In particular, is the following argument outline valid: The Laplace transforms (with $s$ now in $\mathbb{C}$), $s\mapsto \mathbb{E}(e^{sX})$ exist on some strip $\text{Re}(s)\in (-\delta,0)$ or $\text{Re}(s)\in (0,\delta)$ and are analytic there. Therefore, they agree on that strip, and so they agree on the boundary $\text{Re}(s)=0$, so the characteristic functions are the same. That implies the distributions are the same.
Best Answer
Yes it is valid, see Theorem 2 in A note on moment generating functions by Mukherjea, Rao, and Suen.
For convenience I will quote the theorem here:
Let $0<a<b$, $M_n(t)=Ee^{tX_n}$ and $M(t)=E(e^{tX})$ such that $\lim_{n\to\infty} M_n(t)=M(t)$ whenever $a<t<b$. Then $F_n$, the cumulative distribution function of $X_n$, converges weakly to $F$, the cumulative distribution function of $X$.
This theorem includes your case as well, since you can just take $X_n$ to be a constant sequence.
Link to paper: https://www.sciencedirect.com/science/article/abs/pii/S016771520500475X