apply Paley-Wiener to $\frac{\sin z}{z}$ is it true that
$$ \left| \frac{\sin z}{z} \right| \leq C e^{A|z|}$$
for every complex number $z$ ?
is $A=2$ minimal?

it was trivial indeed: using
$$ \sin z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!},$$
dividing and taking abs, one arrives at
$$ \left| \frac{\sin z}{z} \right| \leq \exp |z|.$$
