Let Let \(f:{\mathbb C}\rightarrow {\mathbb C}\) be an entire function. Assume the existence of a non-negative integer \(m\), and of positive constants \(L\) and \(R\), such that for all \(z\) with \(|z|>R\) the inequality \[|f(z)| \leq L |z|^m\] holds. Prove that \(f\) is a polynomial of degree \(\leq m\).
Let \(f:{\mathbb C}\rightarrow {\mathbb C}\) be an entire function. Suppose that there exists a real number M such that for all \(z\in {\mathbb C}\) \[\mbox{\textrm Re} (f) \leq M.\] Prove that \(f\) must be a constant.
Solution
Goal: (a) If an entire \(f\) satisfies \(\abs{f(z)} \leq L\abs z^m\) for \(\abs z > R\), prove \(f\) is a polynomial of degree \(\leq m\); (b) if \(\Re f \leq M\) everywhere, prove \(f\) is constant.
1. (a): For every \(n > m\), \(f^{(n)}(0) = 0\).
Fix \(\rho > R\).
By the Cauchy estimates on \(\abs z = \rho\), \(\abs{f^{(n)}(0)} \leq \frac{n!}{\rho^n} \max_{\abs z = \rho}\abs{f(z)} \leq \frac{n! L \rho^m}{\rho^n} = n! L \rho^{m-n}\). For \(n > m\), \(\rho^{m-n} \to 0\) as \(\rho \to \infty\), so \(f^{(n)}(0) = 0\).
2. (a): \(f\) is a polynomial of degree \(\leq m\).
The Taylor expansion of the entire function \(f\) about \(0\) is \(f(z) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}z^n\); by 1 all terms with \(n > m\) vanish.
3. (b): \(g(z) \definedas e^{f(z)}\) is entire and bounded by \(e^M\).
\(g\) is entire (composition of entire functions), and \(\abs{g(z)} = e^{\Re f(z)} \leq e^M\) by hypothesis.
4. (b): \(g\) is constant, hence \(f\) is constant.
By Liouville’s theorem, the bounded entire function \(g\) of 3 is constant; then \(f = \log g\) is locally constant, and since \(\CC\) is connected, \(f\) is constant.
5. Q.E.D.
2 proves (a) and 4 proves (b).