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
where
.
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
exists?