On the Laplace transform of perpetuities with thin tails

We consider the random variables $R$ which are solutions of the distributional equation $R\overset{\cL}{=}MR+Q$, where $(Q,M)$ is independent of $R$ and $\ABS{M}\leq 1$. Goldie and Gr\"ubel showed that the tails of $R$ are no heavier than exponential. In this note we provide the exact lower and upper bounds of the domain of the Laplace transform of $R$.