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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2019 • Question 9

Question

Math 6210 • Real Analysis Spring 2019 • Question 9

Show that if \(f:\mathbb D\to \mathbb C\) is holomorphic and \(f'(0)\ne 0\) then there is a neighborhood of \(0\) where \(f\) is injective. Do not use the inverse function theorem.

Raw LaTeX

Show that if \(f:\mathbb D\to\mathbb C\) is holomorphic and
\(f'(0)\ne0\) then there is a neighborhood of \(0\) where \(f\) is injective. Do
not use the inverse function theorem.

Topic Tags

Automorphism Groups and Biholomorphisms Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem