Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2015 • Question 9

Question

Math 6210 • Real Analysis Fall 2015 • Question 9

Let \(\mathcal F\) be a family of holomorphic functions on \(\mathbb D\) so that there exists \(C\) with \(|f(z)|<C\) for all \(z\in \mathbb D\).

(a) If \(D\subset \mathbb D\) satisfies that the closure \(\overline D\) is contained in \(\mathbb D\), then \(\mathcal F\) is uniformly Lipschitz on \(D\). That is, there exists \(u\) so that for any \(f\in \mathcal F\) and \(w,z\in D\) we have \[ |f(z)-f(w)|<u|z-w|. \]

(b) Show that \(\mathcal F\) is not necessarily a uniformly Lipschitz family on \(\mathbb D\).

Let \(\mathcal F\) be a family of holomorphic functions on
\(\mathbb D\) so that there exists \(C\) with \(|f(z)|<C\) for all
\(z\in\mathbb D\).

(a) If \(D\subset\mathbb D\) satisfies that the closure \(\overline D\) is
contained in \(\mathbb D\), then \(\mathcal F\) is uniformly Lipschitz on \(D\).
That is, there exists \(u\) so that
for any \(f\in\mathcal F\) and \(w,z\in D\) we have
\[
|f(z)-f(w)|<u|z-w|.
\]

(b) Show that \(\mathcal F\) is not necessarily a uniformly Lipschitz family on
\(\mathbb D\).

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