Problem 7

Exercise 1.7 (Liouville's Theorem and a Characterization of Polynomials)  

1. State Liouville's theorem.
2. Suppose $f$ is entire and there exists $C>0$ and $p\in\mathbb{N}$ such that $\vert f(z)\vert\leq C\vert z\vert^p$ for all $\vert z\vert\geq 1$. Then $f$ is a polynomial.

For (a):

Theorem 1 (Liouville's theorem)   A bounded entire function is constant.

Now for (b):

Proof. As in the proof of the Liouville theorem, $f$ is entire so it has a power series

$$f(z) = \sum_{k=0}^\infty a_k z^k$$ (1.53)

where the coefficients are given by Cauchy's differentiation formula

$$a_k = \frac{f^{(k)}(0)}{k!} = \frac{1}{2\pi i}\int_\gamma\frac{f(\xi)}{\xi^{k+1}}d\xi$$ (1.54)

If $k>p$, then $k+1-p>1$, proving that

$$\left\vert\int_\gamma \frac{f(\xi)}{\xi^{k+1}}d\xi\right\vert \le... ...k+1}}d\xi \leq C\int_\gamma \frac{\vert\xi\vert^p}{\vert\xi\vert^{k+1}}d\xi = 0$$ (1.55)

Therefore, $a_k=0$ for $k>p$, so $f$ is a polynomial of degree at most $p$. $\qedsymbol$