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.

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