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Homework Help: Complex Analysis: Inverse function is holomorphic

  1. Apr 29, 2010 #1
    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].
     
  2. jcsd
  3. Apr 29, 2010 #2
    You're off to a good start. I'd suggest also defining [itex]z_0\equiv f^{-1}(w_0)[/itex] and then using that along with the other defined variables to write out the definitions, in terms of [itex]\epsilon[/itex]s and [itex]\delta[/itex]s, of f being differentiable, and f' and g' being continuous.
     
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