\alpha is meant to be theta.
Suppose that f is an entire function, and there exists a nonnegative number \alpha such that
|f(z)| <= \alpha|z|^n
for all z in C. Show that f is a polynomial of degree at most n. (Hint: Use the Cauchy integral formula to show that f^(n+1)(w)=0 for all w in C.)
Thank you!
Yes, that looks about right. It shows that by choosing R large enough, you can make f^{n+1}(w) arbitrarily small. But since f^{n+1}(w) is independent of R, it follows that it must be 0. If that holds for all w, then f has to be a polynomial of degree at most n.