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!
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1What can you say about $\frac{1}{f}$? – carmichael561 May 06 '16 at 19:52
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That 1/f > 1/M ? – maths.gal May 06 '16 at 19:55
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$1/|f|<1/M$, right? – carmichael561 May 06 '16 at 19:55
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Hint : Apply Liouville theorem to $\frac{1}{f}$.
Bérénice
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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 :). – Bérénice May 06 '16 at 19:59