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Maximum Modulus Theorem

I do understand basic topology, but the proof mentioned here is out of my understanding capacity. Though this could be elementary, I need to know this for further study of Riemann Zeta Functions. This image is taken from 'Lectures on The RZF' by K. Chandrasekharan. Any descriptive clarification will be a great help. Thank you

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  • $\begingroup$ Think to your set $G_1$ as $D$ minus a few points. If $h$ is continuous and $U$ is open then $h^{-1}(U)$ is open (take $h(x_0) \in U$ and find a small neighborhood $V \ni x_0$ such that $h(V) \subset U$). The open mapping theorem is about $h(V)$ being open when $h$ is holomorphic, with the maximum modulus principle it is easy once you know holomorphic implies analytic so that locally $f(z) \approx f(z_0)+ C (z-z_0)^n$ which doesn't have local maximum nor minimum $\endgroup$
    – reuns
    Oct 3, 2018 at 19:12
  • $\begingroup$ $M=\max_{z\in D}|f|?$ Why should the maximum exist? $\endgroup$
    – zhw.
    Oct 3, 2018 at 21:38

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