Does Liouville's theorem hold if the function is bounded from below?

Question

Given f an entire function, assume there exists an m>0 s.t. |f(z)|$\geq$ m $\forall$ z $\in$ $\mathbb{C}$. Show that f is constant. It seems awfully a lot like liouville's theorem but I am not sure on the bound.

Note that you need $m > 0$. – Mees de Vries Feb 05 '19 at 13:21 — Mees de Vries, Feb 05 '19 at 13:21
Apologies missed it out by accident. – DMj Feb 05 '19 at 13:38 — DMj, Feb 05 '19 at 13:38

score 3 · Answer 1 · answered Feb 05 '19 at 13:21

3

Consider $\frac{1}{f}$.

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answered Feb 05 '19 at 13:21

Klaus

10,578

score 0 · Answer 2 · answered Feb 05 '19 at 13:40

As in the proof of the fundamental theorem of algebra using Liouville's theorem, where you assume the polynomial is never zero, then get it bounded away from zero, then invert it... consider the reciprocal of $f$ (if $m\gt0$).

Does Liouville's theorem hold if the function is bounded from below?

2 Answers2