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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2010 • Question 10

Question

Math 6210 • Real Analysis Spring 2010 • Question 10

Let \(f(z)\) be an analytic function in the unit disc \(\{z\mid |z|<1\}\) such that \(f(z)\) is bounded in modulus by \(M\): \(|f(z)|\le M\). Let \(0<r<1\). Find a constant \(C\) depending on \(M\) and \(r\) so that for all \(z,w\) of modulus less than \(r\), \[ |f(z)-f(w)|\le C|z-w|. \]

Raw LaTeX

Let \(f(z)\) be an analytic function in the unit disc
\(\{z\mid |z|<1\}\) such that \(f(z)\) is bounded in modulus by \(M\):
\(|f(z)|\le M\). Let \(0<r<1\). Find a constant \(C\) depending on \(M\) and \(r\) so
that for all \(z,w\) of modulus less than \(r\),
\[
|f(z)-f(w)|\le C|z-w|.
\]

Topic Tags

Analytic Continuation and Monodromy Holomorphic Functions and Cauchy-Riemann Equations