Does there exist such analytic function?

Question

Does there exist a non constant analytic function $f:\bf C \to \bf C$ such that $f(0)=1$ and for $z \in \bf C$ such that $\vert z\vert \geq 1$,we have $$\vert f(z) \vert \leq e^{-\vert z\vert}$$

Any Hints?

Answer 1

If $\lvert f(z)\rvert \leq e^{-|z|}$ for $|z|\geq 1$, then $f$ is clearly bounded outside the unit disk. If $f$ is analytic, then it is clearly bounded inside the unit disk. Thus $f$ would necessarily be a bounded function.

There does not exist a nonconstant bounded analytic function on $\mathbb{C}$.

Answer 2

No, $f$ would violate the maximum principle.