Liouville's Theorem (Complex Analysis)/Corollary
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Corollary to Liouville's Theorem (Complex Analysis)
Let $f: \C \to \C$ be an entire function such that:
- $\size {\map f z} \ge M$
for all $z \in \C$ for some real constant $M > 0$.
Then $f$ is constant.
Proof
Note that since:
- $\size {\map f z} \ge M > 0$
we cannot have:
- $\map f {z_0} = 0$
for any $z_0 \in \C$.
So, by the Quotient Rule for Complex Derivatives, we have:
- $\dfrac 1 f$ is entire.
We also have that:
- $\ds \size {\frac 1 {\map f z} } \le \frac 1 M$
for all $z \in \C$.
So:
So, by Liouville's Theorem (Complex Analysis), we have:
- $\dfrac 1 f$ is constant.
That is, there exists some $C \in \C \setminus \set 0$ such that:
- $\dfrac 1 {\map f z} = C$
for all $z \in \C$.
Then, we have:
- $\map f z = \dfrac 1 C$
for all $z \in \C$.
So $f$ is constant.
$\blacksquare$