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I have the following problem which I don't understand.

Find all entire functions $f$ such that $$|f(z)|\ge \frac{1}{1+|z|^{2017}}=g(z), \; \forall z \in \Bbb{C}$$

The answer says that $f$ has to be constant. I feel like I should use Liouville's theorem. I have tried introducing a function $h(z) = \frac{g(z)}{f(z)}$. But I feel like this is wrong since $g(z)$ is not holomorphic so I know that $h$ is bounded by 1 but nothing else. How should I continue?

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1 Answer 1

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It follows from your condition that $f$ has no zeros. Therefore, $\frac1f$ is also an entire function and$$(\forall z\in\mathbb{C}):\left|\frac1{f(z)}\right|\leqslant1+|z|^{2017}.$$So, $\frac1f$ is a polynomial function whose degree is, at most, $2017$. On the other hand, $\frac1f$ has no zeros. It follows that $\frac1f$ is constant and so $f$ is constant too.

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