# Complex Analysis: Inverse function is holomorphic

1. Apr 29, 2010

### masterslave

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 $$f\prime\neq 0\forall z\in G$$. 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 $$w_{0}\in f(G)$$, and with h small, look at $$\frac{(g(w_{0}+h)-g(w_{0}))}{h}$$.

Because $$f(G)$$ is open, we know there is a $$\delta>0\ni \left|h\right|<\delta$$ gives us $$w_{0}+h\in f(G)$$.
So with h that small, we have $$w_{0}+h=f(z)$$ for some $$z \in G$$.

2. Apr 29, 2010

### foxjwill

You're off to a good start. I'd suggest also defining $z_0\equiv f^{-1}(w_0)$ and then using that along with the other defined variables to write out the definitions, in terms of $\epsilon$s and $\delta$s, of f being differentiable, and f' and g' being continuous.