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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2016 • Question 7

Question

Math 6210 • Real Analysis Spring 2016 • Question 7

Let \(\mathbb D\) be the unit disk \(\{z\in \mathbb C\mid |z|<1\}\). Let \(f\) be a holomorphic function on \(\mathbb D\setminus \{0\}\) such that \[ |f(z)|\le \log \left (\frac {1}{|z|}\right ) \quad \text {for all }z\in \mathbb D\setminus \{0\}. \] Show that \(f=0\).

Raw LaTeX

Let \(\mathbb D\) be the unit disk
\(\{z\in\mathbb C\mid |z|<1\}\). Let \(f\) be a holomorphic function on
\(\mathbb D\setminus\{0\}\) such that
\[
|f(z)|\le \log\left(\frac{1}{|z|}\right)
\quad\text{for all }z\in\mathbb D\setminus\{0\}.
\]
Show that \(f=0\).

Topic Tags

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