Problem: Let \( f(z) \) be an entire function such that \( |f(z)| \leq M |z|^n \) for some positive constants \( M \) and \( n \), and for all \( z \in \mathbb{C} \). Prove that \( f(z) \) is a polynomial of degree at most \( n \).
Solution: Since \( f(z) \) is entire, it can be represented by its Taylor series expansion around \( z = 0 \):
\[
f(z) = \sum_{k=0}^{\infty} a_k z^k, \quad a_k = \frac{f^{(k)}(0)}{k!}
\]
By Cauchy's inequality, for any \( R > 0 \), we have:
\[
\left\vert a_k \right\vert \leq \frac{M R^n}{R^k} = M R^{n - k}.
\]
Letting \( R \to \infty \), we see that \( |a_k| \leq 0 \) for \( k > n \). Therefore, \( a_k = 0 \) for all \( k > n \), and \( f(z) \) reduces to a polynomial of degree at most \( n \)
\[
f(z) = \sum_{k=0}^{n} a_k z^k.
\]