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

Example 6.2.2
Let
. The function
is the composition of two entire
functions, therefore is entire.

This function is obviously non constant. Actually, if
Im
, then
is not bounded.

Theorem 6.2.3 (Generalization of Liouville's theorem)
Let
be an entire function verifying the following property: for
some integer
, there exist two constants
and
such that
. Then
is a polynomial of degree at most
.

Example 6.2.4
Suppose that
is entire and that
. Then
is a polynomial of degree
at most 1.