Suppose we have two functions $f, g: \mathbb{C}^n \rightarrow \mathbb{C}$ in the Paley-Wiener space, i.e. holomorphic functions satisfying $$ |f(\xi)| \leq A(1 + ||\xi||)^N e^{B \text{Im}(\xi)},$$ with some constants $A, N, B$, and same for $g$. I want to show that if $fg \equiv 1$ then $f$ is the Fourier transform of a Dirac delta distribution. The sketch of the proof I've seen starts with saying that clearly $f$ is nowhere zero, hence we may define its logarithm $h$ as a holomorphic function s.t. $f(\xi) = e^{h(\xi)}$. Now the claim is that since $f, g$ satisfy the Paley-Wiener bounds one can get corresponding bounds on $h$, i.e. some constants $c_1, c_2$ s.t. $$ |h(\xi)| \leq c_1 + c_2| \text{Im} (\xi)| \ldotp$$
What puzzles me is that in some sense you get rid of the $(1 + ||\xi||)^N$ term in the estimate on $f$, in particular $f$ needs to be bounded on the real line. I reckon that must use the fact that $f$ is invertible with inverse in the PW space as well, but I haven't been able to figure a reasonable way to use that.