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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2001 • Question 7

Question

Math 6210 • Real Analysis 2001 • Question 7

Let \(f\) be a holomorphic function defined on a neighbourhood of the closed disc \(|z|\le 1\). Assume that \(f(0)=1\), and \(|f(z)|>1\) if \(|z|=1\). Must \(f\) have a zero in the open disc \(|z|<1\)? Justify your answer.

Raw LaTeX

Let \(f\) be a holomorphic function defined on a neighbourhood of the
closed disc \(|z|\le 1\). Assume that \(f(0)=1\), and \(|f(z)|>1\) if \(|z|=1\). Must
\(f\) have a zero in the open disc \(|z|<1\)? Justify your answer.

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Open Mapping and Closed Graph Theorems Zeros of Holomorphic Functions and the Identity Theorem