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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2018 • Question 8

Question

Math 6210 • Real Analysis Fall 2018 • Question 8

Prove or disprove: Let \[ \Omega =\{z:\ |\operatorname {Re}(z)|<1\text { and }|\operatorname {Im}(z)|<1\}. \] If \(f:\mathbb D\to \Omega \) is a biholomorphism so that \(f(z)\) is real for all real \(z\in \mathbb D\) and \(f(0)=0\), then \[ f(iz)=i f(z). \]

Raw LaTeX

Prove or disprove: Let
\[
\Omega=\{z:\ |\operatorname{Re}(z)|<1\text{ and }|\operatorname{Im}(z)|<1\}.
\]
If \(f:\mathbb D\to\Omega\) is a biholomorphism so that \(f(z)\) is real for all
real \(z\in\mathbb D\) and \(f(0)=0\), then
\[
f(iz)=i f(z).
\]

Topic Tags

Automorphism Groups and Biholomorphisms Maximum Modulus Principle