Prove that if \(f\) is an entire function and \(|f(z)|\le a+b|z|^k\) for all \(z\in\mathbb{C}\) where \(a\), \(b\) and \(k\) are positive real numbers, then \(f\) is a polynomial.
Solution
Write \(f(z)=\sum_{m\ge0}c_mz^m\), the Taylor series of \(f\) at \(0\), which converges on \(\CC\).
1. \(\abs{c_m}\le aR^{-m}+bR^{k-m}\) for every \(R>0\).
Cauchy’s estimate on \(\abs z=R\) with the hypothesis \(\abs{f(z)}\le a+bR^k\) there.
2. Q.E.D.
For an integer \(m>k\), letting \(R\to\infty\) in step 1 gives \(c_m=0\). So \(f(z)=\sum_{m\le k}c_mz^m\) is a polynomial of degree at most \(\lfloor k\rfloor\).