TITLE: Applications of the Small and Great Theorems of Picard
QUESTION [8 upvotes]: I just presented the two famous theorems of Picard (Small and Great) in a graduate course, but I have not managed to discover a good number of interesting applications. 
List of applications (rather straight-forward though) found so far: 

If a meromorphic function on $\mathbb C$ misses three values, then it is constant.
The equation $f^n+g^n=1$ has NO non-trivial meromorphic in $\mathbb C$ solutions if $n\ge 3$. 
If $f$ is entire an 1-1, then it is linear.
If $f,g$ are entire and $g'=f(g)$, then $f$ is linear or $g$ is constant.

Could you provide any interesting applications of these theorem?  
I have asked this question in Mathematics StackExchange, but I only received one response.

REPLY [4 votes]: In general, holomorphic maps $f: \mathbb{C} \rightarrow \mathbb{C}$ have no fixed points; but using Picard theorem we can show that $f\circ f$ always have fixed point.
Theorem
(Fixed-point theorem) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic. Then $f \circ f: \mathbb{C} \rightarrow \mathbb{C}$ always has a fixed point unless $f$ is a translation $z \mapsto z+b, b \neq 0 .$
Proof:
Suppose $f \circ f$ has no fixed points. Then $f$ also has no fixed points, and it follows that
$$
g(z):=\frac{f(f(z))-z}{f(z)-z}
$$
is entire function. This function omits the values 0 and 1 ; hence, by Picard, there exists a $c \in \mathbb{C} \backslash\{0,1\}$ with
$$
f(f(z))-z=c(f(z)-z), \quad z \in \mathbb{C}
$$
Differentiation gives $f^{\prime}(z)\left[f^{\prime}(f(z))-c\right]=1-c$. Since $c \neq 1, f^{\prime}$ has no zeros and $f^{\prime}(f(z))$ is never equal to $c$. Thus $f^{\prime} \circ f$ omits the values 0 and $c \neq 0$ by Picard, $f^{\prime} \circ f$ is therefore constant. It follows that $f^{\prime}=$ constant, hence that $f(z)=a z+b$. Since $f$ has no fixed points, $a=1$ and $b \neq 0$