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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

1999 • Question 8

Question

Math 6210 • Real Analysis 1999 • Question 8

Let \(f\) be a non-vanishing holomorphic function defined in a neighborhood of the closed unit disc \(\overline U\) such that \(|f(z)|=1\) if \(|z|=1\). Show that \(f\) is constant.

Raw LaTeX

Let \(f\) be a non-vanishing holomorphic function defined in a
neighborhood of the closed unit disc \(\overline U\) such that \(|f(z)|=1\) if
\(|z|=1\). Show that \(f\) is constant.

Topic Tags

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