The classical Liouville theorem says that
Theorem (Liouville). Every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all in is constant.
The aim of this entry to is generalize the constant , precisely, what happen if we replace by a polynomial?
Followed by this topic we can prove the following theorem.
Theorem (Generalized Liouville). Assume is an entire function. If for all with some positive numbers , then is a polynomial of degree bounded by .
Proof. Denote the Laurent expansion of by
Since is entire, all coefficients with are zero, i.e.
For any integers , one has
Letting we obtain that which implies that is a polynomial of degree at most .
Corollary. For a given entire function , if the following limit
exists then is a polynomial of degree at most .
Question. What happen if