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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1996 • Question 8

Question

Math 6210 • Real Analysis Fall 1996 • Question 8

Let \(D\) be an open set in \(\mathbb C\). Let \(\mathcal F\) be a collection of holomorphic functions defined on \(D\).

(a) Assume \(\mathcal F\) is uniformly bounded on the boundary \(\partial \Delta \) of each triangle \(\Delta \) contained in \(D\). Show that \(\mathcal F\) forms a normal family.

(b) Is \(\mathcal F\) a normal family if we assume it is uniformly bounded when restricted to any interval contained in \(D\)?

Raw LaTeX

Let \(D\) be an open set in \(\mathbb C\). Let \(\mathcal F\) be a
collection of holomorphic functions defined on \(D\).

(a) Assume \(\mathcal F\) is uniformly bounded on the boundary \(\partial\Delta\) of
each triangle \(\Delta\) contained in \(D\). Show that \(\mathcal F\) forms a normal
family.

(b) Is \(\mathcal F\) a normal family if we assume it is uniformly bounded when
restricted to any interval contained in \(D\)?

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Normal Families and Montel's Theorem Zeros of Holomorphic Functions and the Identity Theorem