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I would be glad to get some help with this question:

Let $f(z)$ be an entire function. Assume that there exists a monotonous increasing and unbounded sequence $\{r_n\}$ such that $\lim\limits_{n \to \infty} \min\limits_{|z|=r_n} |f(z)|=\infty$. I want to show that there exists a $z_0 \in \mathbb C$ that satisfies $f(z_0)=0$.

I'd especially like to know how to use that fact about the sequence.

Thanks.

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    kahen - I ment sequence, I'll edit the question, thanks. GEdgar - Yes, I know the maximum modulus pronciple, but I don't know how to use it here.2011-12-12

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Assume $f(z)\ne0$ for all $z\in\mathbb{C}$. Then $h(z)=1/f(z)$ is also an entire function. Apply the maximum modulus principle to $h$.

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    No. The limit of the maxima in the circles $|z|=r_n$ is 0. In particular $h$ is bounded...2011-12-13