Exploring Bijective Holomorphic Maps on Open Sets and the Unit Circle

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Homework Statement

Let $$f$$ and $$g$$ be bijective holomorphic maps from an open set $$A$$ to the unit circle. Let $$a \in A$$ and $$c=f(a)$$ and $$d=g(a)$$. Find a relation between $$f$$ and $$g$$ that involves $$a,c,d,f'(a),g'(a)$$.

Homework Equations

The Attempt at a Solution

If we also assumed that the open set is connected and simply connected, then we could apply the Riemann Mapping Theorem.
If that were the case, then there is a bijective holomorphic map $$f^{-1}: \text{unit circle} \rightarrow A$$ and $$g^{-1}: \text{unit circle} \rightarrow A$$ Then by the open mapping theorem, $$f: A \rightarrow \text{unit circle}$$ is continuous, thus a homeomorphism between $$A$$ and the unit circle.

I'm not sure where to start because we don't have that "connected and simply connected" assumption.

 

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We also know that $$f,g$$ are biholomophic which implies that it's a conformal map, thus it is angle preserving. Is this a way to two conformal mapping to each other?