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Suppose $f(z)$ is analytic at $z_{0}$ with $f^{\prime}\left(z_{0}\right) \neq 0 .$ Show that there exists an analytic function $g(z)$ such that $f(g(z))=z$ in some neighborhood of $z_{0}$.

This is also known as inverse function theorem. I am looking for the analytical proof that does not use neither Jacobian nor inverse function theorem for $R^2$.

I have tried to use Open Mapping Theorem, but couldn't prove that mapping is bijective. Any help or tips?

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