Let \(f\) be an entire function with \(|f(z)| \leq 3\log|z|\) when \(|z| > 2\). Either verify that \(f\) must be constant, or give a counterexample.
Solution
\(f\) must be constant. The admissible functions are exactly the constants \(c\) with \(|c|\leq3\log2\).
1. Every Taylor coefficient of \(f\) of order \(n\geq1\) vanishes.
Write \(f(z)=\sum_{n\geq0}a_nz^n\) on \(\mathbb C\). For \(R>2\), Cauchy’s estimate on the circle \(|z|=R\) gives \[ |a_n|\leq\frac{3\log R}{R^n} \] [@SS03]. For \(n\geq1\) the right-hand side tends to zero as \(R\to\infty\), so \(a_n=0\). Hence \(f\equiv a_0\) is constant.
2. A constant \(c\) satisfies the bound exactly when \(|c|\leq3\log2\).
The bound requires \(|c|\leq3\log|z|\) for every \(|z|>2\). The infimum of the right-hand side over \(|z|>2\) is \(3\log2\), approached as \(|z|\downarrow2\). Thus the bound holds exactly when \(|c|\leq3\log2\).