- If \(f = u + i v\) is an entire function with the property that \(u ( z ) \leq 0\) for all \(z \in \mathbb { C }\) , what can you say about \(f ?\)
Solution
1. \(e^f\) is a bounded entire function.
\(e^f\) is entire, and \(\abs{e^{f}}=e^u\le1\).
2. Q.E.D.
By step 1 and Liouville’s theorem, \(e^f\) is constant, so \(0=(e^f)'=f'e^f\). Since \(e^f\) never vanishes, \(f'\equiv0\), and \(\boxed{f\text{ is a constant with real part}\le0}\). Every such constant satisfies the hypothesis.