Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an entire function. if there exists $\delta> 0$ and $w\in \mathbb{C}$ such that
$$\left | f(z)-w \right | \geq \delta \qquad \forall z\in\mathbb C $$
Prove that $f$ is constant.
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Let $g(z) = \frac{1}{f(z)-w}$, then $g$ is entire and $|g(z)| \le \delta$ for all $z$. What can you say about $g$ and hence $f$?
copper.hat
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are entire and bounded? – carpediemfg Oct 30 '13 at 17:32
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Well, what can you say about a bounded entire function? – copper.hat Oct 30 '13 at 17:35
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umm, I apply the Liouville theorem and according to that, then g is constant, then f is constant, right? – carpediemfg Oct 30 '13 at 17:43
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Well, first conclude that $g$ is constant, then the formula for $g$ shows that $f$ is constant. – copper.hat Oct 30 '13 at 17:44
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ok yes, thank you very much for your help – carpediemfg Oct 30 '13 at 17:49
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@user99909: Glad to help. – copper.hat Oct 30 '13 at 17:53