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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2024 • Question 3

Question

Math 6220 • Complex Analysis Fall 2024 • Question 3

Let \(f\) be an entire function with \(f(0)=0\). Define

\[ M_f(r):=\sup \{|f(z)|: |z|=r\}. \]

Show that \(M_f(r)\) is non-decreasing. What can you say about \(f\) if it is not strictly increasing?

Raw LaTeX

Let \(f\) be an entire function with \(f(0)=0\). Define

\[
M_f(r):=\sup\{|f(z)|: |z|=r\}.
\]


Show that \(M_f(r)\) is non-decreasing. What can you say about \(f\) if it is
not strictly increasing?

Topic Tags

Entire Functions and Liouville-Type Theorems