Liouville's Theorem.

Theorem 6.2.1   Let $f$ be an entire function. If $f$ is bounded, then $f$ is a constant function.

Example 6.2.2   Let $f(z)=e^{iz}$ . The function $f$ is the composition of two entire functions, therefore is entire.

This function is obviously non constant. Actually, if Im$(z)<0$ , then $f(z)$ is not bounded.

Theorem 6.2.3 (Generalization of Liouville's theorem)   Let $f$ be an entire function verifying the following property: for some integer $k \geq 0$ , there exist two constants $\alpha$ and $beta$ such that $\forall z \in \mathbb{C}, \vert f(z)\vert \leq \alpha + \beta \vert z\vert^k$ . Then $f(z)$ is a polynomial of degree at most $k$ .

Example 6.2.4   Suppose that $f$ is entire and that $\forall z \in \mathbb{C}, \vert f(z)\vert \leq \alpha + \beta \vert z\vert^{4/3}$ . Then $f(z)$ is a polynomial of degree at most 1.