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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 10

Question

Math 6210 • Real Analysis Spring 2008 • Question 10

Let \(\Omega \) be an open subset of \(\mathbb C\) and \(f:\Omega \to \mathbb C\) a holomorphic function. Let \(z\) be a point in \(\Omega \) and assume that \(\Omega \) contains a disk \(D\) centered at \(z\) of radius \(R\). Also assume that \(f(D)\) is contained in a disk \(D'\) centered at \(f(z)\) of radius \(r\). Show that \(|f'(z)|\le r/R\).

Raw LaTeX

Let \(\Omega\) be an open subset of \(\mathbb C\) and
\(f:\Omega\to\mathbb C\) a holomorphic function. Let \(z\) be a point in \(\Omega\)
and assume that \(\Omega\) contains a disk \(D\) centered at \(z\) of radius \(R\).
Also assume that \(f(D)\) is contained in a disk \(D'\) centered at \(f(z)\) of
radius \(r\). Show that \(|f'(z)|\le r/R\).

Topic Tags

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