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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 9

Question

Math 6210 • Real Analysis Fall 2008 • Question 9

Let \(\Omega \subsetneq \mathbb C\) be a simply connected region, and fix \(z_0\in \Omega \). If \(\phi :\Omega \to D\) is a conformal map such that \(\phi (z_0)=0\), show that \[ |\phi '(z_0)| =\sup \{|f'(z_0)|:\ f:\Omega \to D\text { holomorphic}, f(z_0)=0\}. \]

Raw LaTeX

Let \(\Omega\subsetneq\mathbb C\) be a simply connected region, and
fix \(z_0\in\Omega\). If \(\phi:\Omega\to D\) is a conformal map such that
\(\phi(z_0)=0\), show that
\[
|\phi'(z_0)|
=\sup\{|f'(z_0)|:\ f:\Omega\to D\text{ holomorphic}, f(z_0)=0\}.
\]

Topic Tags

Conformal Moduli and Annuli Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains