Let $f$ be a non-constant holomorphic function in a domain $U$. Assume $z_0 ∈ U$ is a local minimum of $\lvert f\rvert$. Show that $f(z_0) = 0$. I have a question from my homework:

Let $f$ be a non-constant holomorphic function in a domain $U$. Assume $z_0 \in U$ is a local minimum of $|f|$. Show that $f(z_0) = 0$.

I know the maximum principle, but I can't figure out the minimum. I thought about Cauchy's integral for sub domain of $U$, but am unsure if that's any useful. 

Recall that $f(U)$ is open.
If $f(z_0)\neq 0$ draw the arrow from $0$ to $f(z_0)$ .
Don't compute anything, just look at the contradiction!
