- (10 points) Let \(f : \mathbb { C } \to \mathbb { C }\) be an entire function. Prove that if there exists some real number C and some positive integer k so that
\[ | f ( z ) | \leq C | z | ^ { k } \]
for all z with \(| z | > 1\) , then f is a polynomial in z of degree at most \(k .\)
Solution
Write \(f(z)=\sum_{n\ge0}a_nz^n\), the Taylor series of \(f\) at \(0\), which converges on \(\CC\).
1. \(\abs{a_n}\le CR^{k-n}\) for every \(R>1\).
Cauchy’s estimate on \(\abs w=R\) with the hypothesis \(\abs{f(w)}\le CR^k\) there.
2. Q.E.D.
For \(n>k\), letting \(R\to\infty\) in step 1 gives \(a_n=0\), so \(f(z)=\sum_{n=0}^ka_nz^n\) is a polynomial of degree at most \(k\).