1. ## liouville's theorem?

Suppose f is an entire function that satisfies $\displaystyle |f(z)|\leq M(1+|z|^m)$ for some constant M and positive integer m. Show that f is a polynomial of at most degree m.

How should I approach this? Do I try to bound $\displaystyle |\frac{f(z)} {1+z^m}|$, and then apply Liouville's theorem? The expression isn't necessarily entire though.

2. Use Cauchy estimates to show that

$\displaystyle \displaystyle \bigg|\frac{\partial^{m+k} }{\partial z^{m+k}}f(z)\bigg|_{z=0}=0$

This will show that the power series of f about zero has only finitely many terms and is therefore a polynomial.

3. Does this work?

For $\displaystyle 0\leq |z|\leq R$, $\displaystyle f(z)\leq M(1+ R^m)$.

Then I use the Cauchy estimate:

For $\displaystyle k>0$, $\displaystyle |f^{(m+k)}(0)|\leq \frac{(m+k)!M(1+R^m)}{R^{m+k}}$. Then I use the trick from Liouville's theorem and take R to infinity, so $\displaystyle f^{(m+k)}(0)=0$. From the power series at 0, f has at most degree m.

4. Yep that is the idea!