A holomorphic mapping f : U → V is a local bijection on U if for every z ∈ U there exists an open disc D ⊂ U centered at z, so that f : D → f(D) is a bijection.
Prove that a holomorphic map f : U → V is a local bijection on U if and only if f (z) ≠ 0 for all z ∈ U. [Hint: Use Rouch´e's theorem as in the proof of Proposition 1.1.]