Analytic, never zero on domain D graph and its local max. min

Question

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum.

Now,

If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show that $|f|$ has no strict local maximum within its domain of analyticity by contradiction.

My question is, how do I use this to show that $|f(z)|$ has no local minimum?

Also, one conceptual question from my textbook is that, what does it mean when they say that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$) true for "sufficiently" small $r$?

Answer 1

For the conceptual question, we see that for $t\in[0,2\pi]$ and $r$ fixed, then $z+re^{it}$, is a circle radius $r$ about $z$, so for sufficiently small $r$ gives us a local condition.