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Could you please advice me some useful analysis results which might be necessary to be used with Liouville's theorem ("any bounded entire function is constant"), Open mapping theorem ("non-constant holomorphic function is open"), Maximum modulus principle.

For instance, in one of the course exercises I had to use the fact that a continuous function on a compact set achieves a maximum. I would really appreciate if you give an example of application of the result that you bring in connection to

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  • $\begingroup$ The maximum modulus principle is used in an enormous number of results. $\endgroup$ – DanielWainfleet Jul 3 '18 at 5:31
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I think it was James Littlewood who wrote "Could a fellowship be awarded for a dissertation of 2 lines? Presumably. For example:

Theorem. A non-constant complex polynomial $p$ has a zero.

Proof. Otherwise $1/p$ is a non-constant bounded entire function." (END QUOTE).

The Fundamental Theorem of Algebra was actually first proved by Gauss, well before Liouville's theorem (and Littlewood knew this). But it can be easily proved using Liouville's result:

Let $p:\Bbb C\to \Bbb C$ be a non-constant polynomial. Since $|p(z)|\to \infty$ uniformly as $|z|\to \infty$ there exists $r\in \Bbb R^+$ such that $|z|\geq r\implies |p(z)|>|p(0)|.$ Therefore $\inf \{|p(z)|:z\in \Bbb C\}=\inf \{|p(z)|:|z|\leq r\}=$ $=\min \{|p(z)|:|z|\leq r\}.$

This $\min$ must be $0,$ otherwise $1/p$ is a non-constant bounded entire function.

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  • $\begingroup$ $|p(z)|$ is continuous and $S=\{z\in \Bbb C: |z|\leq r\} $ is compact and not empty so $\min \{|p(z)|:z\in S\}$ exists. $\endgroup$ – DanielWainfleet Jul 3 '18 at 5:26
  • $\begingroup$ Thank you, Daniel. Yes, we had this proof of the Fundamental theorem of algebra in the course. $\endgroup$ – Aleksei N Jul 3 '18 at 7:04
  • $\begingroup$ A generalization of the Fundamental Theorem of Algebra: If $U$ is a non-empty connected open subset of $ \Bbb C$ and if $f$ is analytic and non-constant on $U$ then the real-valued function $|f|$ cannot have any local minimum value other than $0.$ $\endgroup$ – DanielWainfleet Aug 28 '18 at 0:49

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