Analyticity and connectedness

Question

If $|f(z)| \geq |f(z_0)| >0$ for any $z$ in the domain $D$ which is open and connected ($f: D \to \mathbb{C}$ is analytic) then $g(z) = 1/f(z)$ is analytic on D and $ |g(z)| \leq |g(z_0)|$ for any $x \in D.$ So, $g$ is constant. My question is, what if $|f(z_0)| =0?$ Is $f$ constant?

Answer 1

If $f$ has a zero at $z_0 \in D$ then $|f(z)| \ge |f(z_0)| = 0$ is trivially satisfied for all $z \in D$. This does not imply that $f$ is constant.

You can not apply the same reasoning since $1/f$ is not analytic in $D$.