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This is the well known definition of conformal equivallence between domains in the complex plane:

Let $U$ and $U^{\prime}$ be two domains in the complex plane. We say $U$ and $U^{\prime}$ are conformally equivalent if there is a map $f$ between them such that $f$ is holomorphic and inyective (a conformal map).

My question is: having one such map between these domains, say $f: U \rightarrow U^{\prime},$ implies the existence of a conformal map in the other direction: $g: U^{\prime} \rightarrow U?$ And if so, what theorem guarantees this?

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Every injective holomorphic map has nowhere vanishing derivative. Hence, it's inverse is also holomorphic.

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