(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The problem is from Sarason, page 44, Exercise IV.14.1.

Let f be a univalent holomorphic function in the open connected set G, and let g be the inverse function.

Assume that f(G) is open, that g is continuous, and that [tex]f\prime\neq 0\forall z\in G[/tex]. Prove g is holomorphic.

2. Relevant equations

3. The attempt at a solution

I've tried 3 different methods, failing with each. I'd appreciate any insight into the problem.

My professor recommended looking at the calculus proof of the derivative of an inverse, and the following:

Start with [tex]w_{0}\in f(G)[/tex], and with h small, look at [tex]\frac{(g(w_{0}+h)-g(w_{0}))}{h}[/tex].

Because [tex]f(G)[/tex] is open, we know there is a [tex]\delta>0\ni \left|h\right|<\delta[/tex] gives us [tex]w_{0}+h\in f(G)[/tex].

So with h that small, we have [tex]w_{0}+h=f(z)[/tex] for some [tex]z \in

G[/tex].

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# Homework Help: Complex Analysis: Inverse function is holomorphic

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