Understanding a small step in the proof of Maximum Modulus Principle in ProofWiki

Asked Sep 26 '22 at 11:28

Active May 20 '23 at 07:07

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Consider the given proof of the Maximum Modulus Principle in ProofWiki. This is really just a minor question, but the given proof says that

However, since this this holds for all sufficiently small $r > 0$, $|f|$ would be constant in $B_r(z)$.

Is the proof trying to say that the supposed equality holds for all $0 < r' \leq r$ by repeated application of the mean value property of holomorphic functions or is there a reason why we'd need to make some special choice for $0 < r' \leq r$?

asked Sep 26 '22 at 11:28

Cartesian Bear

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Where did you get inequalities of r? – tryst with freedom Sep 26 '22 at 11:34
@TrystwithFreedom My understanding was that that once we suppose that $|f(z)| = \max_{\theta \in [0, 2\pi)}|f(z + r\exp(i\theta))|$ for a fixed $r > 0$, we reapply the mean value property for all $0 < r' \leq r$. But I just started to wonder about the exact wording of the proof, namely that does the "sufficiently small" convey the same main idea. – Cartesian Bear Sep 26 '22 at 11:56
Related – tryst with freedom May 20 '23 at 09:45
Fairly certain this proof is wrong, check another resource (5/20/23) – tryst with freedom May 20 '23 at 11:31

Understanding a small step in the proof of Maximum Modulus Principle in ProofWiki

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