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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2000 • Question 9

Question

Math 6210 • Real Analysis 2000 • Question 9

Let \(f\) be a holomorphic function on the unit disc. For every \(0\le r<1\) define \[ M(r)=\max _{|z|=r}|f(z)|. \] Show that \(M(r)\) is an increasing, continuous function on \([0,1)\).

Raw LaTeX

Let \(f\) be a holomorphic function on the unit disc. For every
\(0\le r<1\) define
\[
M(r)=\max_{|z|=r}|f(z)|.
\]
Show that \(M(r)\) is an increasing, continuous function on \([0,1)\).

Topic Tags

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