How do bijective holomorphic maps relate 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?