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
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.