Let $\phi$ be an entire function of exponential type. That is:
$$\exists M, c>0,\quad \forall z\in \mathbb{C}\quad |\phi(z)|\leq M e^{c|z|}. $$
On any compact set $|z|=R$, we have:
$$|\phi(z)|\leq M e^{cR} $$ Hence, $\phi$ is a bounded entire function on any compact set. My question is the following:
Can we apply the theorem of Liouville to say that $\phi$ is contant on any compact? or the Liouville Theorem is only applied on the whole complex plane and so we cannot deduce anything from the previous statement?
Many thank's.
Liouville's Theorem is applicable only to bounded analytic functions on the entire complex plane. Every entire function is bounded on compact sets. For example $f(z)=z$ is also bounded on compact sets but it is not constant on compact sets.