- Prove the following statement: If f and g are entire functions, \(g ( z ) \neq 0\) and \(| f ( z ) | \leq | g ( z ) |\) for all \(z \in \mathbf { C } .\) , then \(f ( z ) = C g ( z )\) for some constant C.
Solution
1. \(h\da f/g\) is entire and \(\abs h\le1\).
\(g\) has no zeros, so \(h\) is holomorphic on \(\CC\), and \(\abs{h}=\abs f/\abs g\le1\) by hypothesis.
2. Q.E.D.
By step 1 and Liouville’s theorem, \(h\) is a constant \(C\), so \(f=Cg\).