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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1995 • Question 6

Question

Math 6210 • Real Analysis Fall 1995 • Question 6

If \(f\) is holomorphic in \(\Delta (0;1)=\{z\in \mathbb C:|z|<1\}\) and \(r<1\) prove that there exists a constant \(M\) so that \[ |f(z)-f(w)|\le M|z-w| \] for all \(z,w\in \Delta (0;r)=\{z\in \mathbb C:|z|<r\}\).

Raw LaTeX

If \(f\) is holomorphic in
\(\Delta(0;1)=\{z\in\mathbb C:|z|<1\}\) and \(r<1\) prove that there exists a
constant \(M\) so that
\[
|f(z)-f(w)|\le M|z-w|
\]
for all \(z,w\in\Delta(0;r)=\{z\in\mathbb C:|z|<r\}\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem