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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2020 • Question 8

Question

Math 6210 • Real Analysis Spring 2020 • Question 8

Show that if \(f\) is an entire function and \(f\) restricted to the unit disk is a bijection, that is, \(f|_{\mathbb D}:\mathbb D\to \mathbb D\) is a bijection, then \(f\) is a rotation. That is, there exists \(c\in \mathbb C\) with \(|c|=1\) so that \(f(z)=cz\) for all \(z\).

Raw LaTeX

Show that if \(f\) is an entire function and \(f\) restricted to the
unit disk is a bijection, that is, \(f|_{\mathbb D}:\mathbb D\to\mathbb D\) is
a bijection, then \(f\) is a rotation. That is, there exists \(c\in\mathbb C\)
with \(|c|=1\) so that \(f(z)=cz\) for all \(z\).

Topic Tags

Automorphism Groups and Biholomorphisms Entire Functions and Liouville-Type Theorems