Determine all entire functions \(f\) (i.e., \(f(z)\) is holomorphic on the whole \(z\)-plane) that satisfy the inequality
\[|f(z)| \leq |z|^2 |\mathrm{Im}\, z|^2\]
for \(|z|\) sufficiently large.
Solution
The only possibility is \(\boxed{f\equiv0}\).
1. The inequality forces zeros on a real interval.
Choose \(R>0\) so that the given bound holds whenever \(|z|>R\). For every real \(x>R\), \[ 0\leq|f(x)|\leq|x|^2|\operatorname{Im}x|^2=0. \] Thus \(f(x)=0\) for all such \(x\). In particular the distinct zeros \(x_n=R+1+1/n\) accumulate at \(R+1\), a point of the domain \(\mathbb C\).
2. The identity theorem and the converse finish the classification.
The entire function \(f\) has zeros accumulating inside the connected plane, so the identity theorem gives \(f\equiv0\) [@SS03]. Conversely, the zero function is entire and satisfies the given inequality at every point, since its right-hand side is nonnegative.