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First, I assume you want $w$ to be holomorphic (or else it isn't true). If the target is really $S^2$ with boundary on the equator $L$ then this should be easy to prove directly. Take a point $p$ not on $L$. We can define an intersection number between $w$ and $p$ using the intersection pairing between $H_2(S^2,L)$ and $H_0(S^2\setminus L)$. By positivity of ...

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