Determine all entire functions \(f: \mathbb{C} \to \mathbb{C}\) for which \(|f(z)| \le |z|^2\) for all \(z \in \mathbb{C}\).
Solution
Write \(f(z)=\sum_{n\ge0}a_nz^n\), the Taylor series of \(f\) at \(0\), which converges on \(\CC\).
1. \(a_n=0\) for \(n\ge3\).
Cauchy’s estimate on \(\abs z=R\) gives \(\abs{a_n}\le R^{-n}\max_{\abs z=R}\abs f\le R^{2-n}\), which tends to \(0\) as \(R\to\infty\) when \(n\ge3\).
2. \(a_0=a_1=0\).
\(\abs{a_0}=\abs{f(0)}\le0\). Then \(\abs{a_1+a_2z}=\abs{f(z)}/\abs z\le\abs z\) for \(z\ne0\), and letting \(z\to0\) gives \(a_1=0\).
3. Q.E.D.
By steps 1 and 2, \(f(z)=a_2z^2\), and the bound at \(z=1\) gives \(\abs{a_2}\le1\). Conversely, \(cz^2\) with \(\abs c\le1\) satisfies the bound. So the functions are \(\boxed{f(z)=cz^2,\ \abs c\le1}\).