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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 9

Question

Math 6210 • Real Analysis Fall 2007 • Question 9

Let \(\Omega \) be an open connected subset of \(\mathbb C\) and \(\mathcal F\) a family of holomorphic functions with domain \(\Omega \) such that for every compact subset \(K\) of \(\Omega \) there is a positive real constant \(M_K\) such that for all \(f\in \mathcal F\) and \(z\in K\) we have \(|f(z)|<M_K\). Show that for every \(z\in \Omega \) there exists a positive constant \(B_z\) such that \(|f'(z)|<B_z\) for all \(f\in \mathcal F\).

Raw LaTeX

Let \(\Omega\) be an open connected subset of \(\mathbb C\) and
\(\mathcal F\) a family of holomorphic functions with domain \(\Omega\) such that
for every compact subset \(K\) of \(\Omega\) there is a positive real constant
\(M_K\) such that for all \(f\in\mathcal F\) and \(z\in K\) we have \(|f(z)|<M_K\).
Show that for every \(z\in\Omega\) there exists a positive constant \(B_z\) such
that \(|f'(z)|<B_z\) for all \(f\in\mathcal F\).

Topic Tags

Compactness in Metric and Normed Spaces Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains