In Brown and Churchill's book, the maximum modulus principle is stated as follows:
If a function $f$ is analytic and not constant in a given domain $\Omega$, then $|f (z)|$ has no maximum value in $\Omega$.
In the proof, they stated the following. The question I have is as $d$ is a distance, wouldn't it always non-negative? How come it is only positive when it's the whole plane ?
Draw a polygonal line $L$ lying in $\Omega$ and extending from $z_0$ to any other point $P$ in $\Omega$. Let $d$ represent the shortest distance from points on $L$ to the boundary of $\Omega$. When $\Omega$ is the entire plane, $d$ may have any positive value.