Is this complex exponential inequality true? Is it true that $|f(z)|\leq M$ implies that $|e^{f(z)}|\leq e^{M}$.  If not, what is a simple counterexample?

For $f=u+iv$ we have $u(z)\le |f(z)|\le M$ and
$$
|e^{f(z)}|=|e^{u(z)}e^{iv(z)}|=e^{u(z)}\le e^M
$$
since $e$ is increasing.
