For $f$ that is analytic on domain D, $|f(z)|$ has no local maximum within its domain of analyticity.
I have proven the result that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), and am trying to use it show that $|f|$ has no strict local maximum within its domain of analyticity by contradiction.
Is it just assuming that if there is a maximum say $f(z_0)$ then we can find a small enough r s.t. $f(z_0+re^{it})$ is still within the domain but bigger than $f(z_0)$? If so, I was wondering how can I write it down in a more rigorous way.
By Maximum Modulus Principle the fact that $f$ attains its maximum at an interior point of some closed disk around $z$ implies that $f$ is a constant on the disk. Since the domain of $f$ is connected this implies that $f$ is a constant throughout $D$.