Today during the qualifying exam I met this question:
Show that the maximum modulus principle implies the Liouville Theorem.
Well, this is my attempt:
It suffices to show that a bounded entire function can achieve its maximum modulus in complex plane. But I got messed up here. Can anyone give me some ideas?
Best Answer
The function
$$f(z) = \frac{z}{1+\lvert z\rvert}$$
which is a homeomorphism between $\mathbb{C}$ and the unit disk shows that the maximum modulus principle alone does not imply Liouville's theorem. We need some more properties of holomorphic functions.
Something closely related: Riemann's removable singularity theorem.
The Riemann sphere is compact, so every continuous function on the entire sphere ...