In how many ways can we prove maximum modulus principle ? We have open mapping theorem by which we can prove that.can we prove the theorem also using the following fact ? Uniqueness theorem: Let $G$ be a region, $f:G \rightarrow \mathbb{C}$ be analytic.Then TFAE: (i) $f=0$ (ii) there is a point $a$ in $G$ such that $f^{(n)} (a)=0$ for each $n \geq 0$ (iii) $\{ z \in G: f(z)=0 \}$ has a limit point in $G$.
proving maximum modulus theorem in different ways
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complex-analysis
complex-numbers
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0It can be directly proven using C-R equation. Moreover, maximum modulus principle can be proven from Gauss mean integral theorem. Knowing this, try to prove that for holomorphic functions $f_1,...,f_n$, $|f_1|+...+|f_n|$ have the maximum modulus property. It was a good exercise to me :) – 2017-01-19
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1But I need to prove it from the following facts as mentioned by me – 2017-01-19
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0Use Gauss mean integral theorem to prove that a nonconstant holomorphic function cannot attain its maximum at a ball in its domain. Then, apply identity theorem to extend the ball to the whole domain. – 2017-01-19