# Show that an entire function is a constant [duplicate]

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Let $f$ be entire and suppose there is a constant $M>0$ such that $|f(z)|>M$ for all $z \in \mathbb C$. Prove that $f$ is constant.

I think this has something to do with Liouville's theorem but not sure how to go about it!

## marked as duplicate by Martin R, amd, zz20s, user228113, colormegoneMay 6 '16 at 22:30

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• What can you say about $\frac{1}{f}$? – carmichael561 May 6 '16 at 19:52
• That 1/f > 1/M ? – maths.gal May 6 '16 at 19:55
• $1/|f|<1/M$, right? – carmichael561 May 6 '16 at 19:55

## 2 Answers

Show that $g(z):=\frac1{f(z)}$ is constant.

Hint : Apply Liouville theorem to $\frac{1}{f}$.

• How do I know to do that though? – maths.gal May 6 '16 at 19:55
• well to apply Liouville theorem your function needs to be entire and it must be bounded. The intuition is that if you have $|f|>M$ then $|\frac{1}{f}|<1/M$. But after yoi have to verify all the other conditions, since $f \neq 0$, $\frac{1}{f}$ is also entire. So the intuitions was true :). – Jennifer May 6 '16 at 19:59