Prove that if a function respects the Cauchy Riemann condition and it is invertible, it is conformal.

Question

I need to prove that if a function respects the Cauchy Riemann condition and it is invertible, it is conformal.

I'm familiar with the proof that if a function is holomorphic and has non vanishing derivate then it's a conformal mapping. But i don't really know how to proof this one.

Answer 1

If a holomorphic function is one-to-one then the derivative never vanishes. Ref. See the theorem just before Global Cuachy's Theorem in Rudin's Real and Complex Analysis.