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I'm solving the following problem.

Let $\Omega = \lbrace z \in \mathbb{C} : -1< \operatorname{Im} z <1 \rbrace$ and $f$ be a holomorphic function from $\Omega$ to the unit disk satisfying its limit to $ \infty$ along real axis is 0. Then prove that for any $-1<y<1$, $f(x+iy)\rightarrow 0$ as $x \rightarrow \infty$

I tried to use a LFT to unit disk to avobe strip so that consider the composition of it yields a map from unit disk to itself to use Schwarz lemma. BUT I found that such LFT does not exist.. IS THERE ANYWAY TO APPROACH THIS? any comment will be appreciated.

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up vote 1 down vote accepted

Here is a nifty proof using (the easy version of) Montel's theorem:

Let $f_n(z) = f(z+n)$. Then each $f_n$ is a holomorphic map from $\Omega$ to $\mathbb{D}$. In particular, the family $(f_n)$ is uniformly bounded in $\Omega$, hence a normal family. This means that any subsequence of $(f_n)$ has a locally uniformly convergent subsequence, converging to some holomorphic limit $f$. By assumption, $f(x) = 0$ for $x \in \mathbb{R}$, so by uniqueness principle $f\equiv 0$. Since all limit functions are $0$, we conclude that $f_n \to 0$ locally uniformly in $\Omega$, which implies the claim.

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I can't for the life of me figure out why $f_n\to 0$. I understand that every sequence has a subsequence $\to 0$. But why is $f_n \to 0$. Thanks. – PeterM Mar 2 '13 at 20:10

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