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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1993 • Question 8

Question

Math 6210 • Real Analysis Fall 1993 • Question 8

Let \(D\) be an open connected set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\), which is uniformly bounded on each circle contained in \(D\). Show that \(\mathcal F\) forms a normal family.

Raw LaTeX

Let \(D\) be an open connected set in \(\mathbb C\). Let \(\mathcal F\) be
a collection of holomorphic functions defined on \(D\), which is uniformly
bounded on each circle contained in \(D\). Show that \(\mathcal F\) forms a normal
family.

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Normal Families and Montel's Theorem Riemann Mapping Theorem and Simply Connected Domains