Conceptual explanation for basic theorem complex analysis

Question

From Fundamentals of Complex Analysis by E.B. Snaff and A.D. Snider,

$\text{Thm}. 23:$ If $f$ is analytic in a domain $D$ and $|f(z)|$ achieves its maximum value at a point $z_0$ in $D$, then $f$ is constant in $D$.

My current interpretation of this theorem: If the modulus of $f(z)$ achieves a maximum at $z_0$, then $|f(z_0)| \geq |f(z_1)|$ where $z_0, z_1\in D$ and $z_0 \neq z_1$. If $f$ is actually constant, then isn't $|f(z_0)|=|f(z)|, \forall z\in D$?

How then did we determine that $|f(z_0)|$ was a maximum in the first place?

Answer 1

The key concept to understand here is the following: if $f(z)$ is complex analytic function and at some point $z_0$ we have $f'(z_0)$ not equals to $0$, then in a small neighborhood of $z_0$ the function $f$ behaves like linear $a + bz, b \neq0$. It is obvious that the linear function can not achieve its maximum value at any inner point in D. Using standard continuity considerations, it is easy to show, that $f$ is then also satisfies the same rule.

Therefore analytic function $f$ may achieve its maximum value at inner point in $D$ IFF it is constant in $D$.

Answer 2

$|f(z_0)| \ge |f(z_1)|$ for all $z_1 \in D$ is just a restatement of "$|f(z)|$ achieves its maximum value at $z_0$". But the theorem says something much stronger: for a non-constant analytic function, this can't happen.

As for your question: if $f$ is constant, then every point in $D$ is a maximum.