Picard's little theorem

Remarks

One classical proof uses the modular lambda function, which gives a holomorphic covering map from the unit disc onto $\C \setminus \set{0, 1}$. If $f$ omits two values, one may assume that they are $0$ and $1$. Then $f$ lifts through this covering to an entire function with values in the unit disc, which is constant by Liouville's theorem, so $f$ is constant. Another proof uses Bloch's theorem. The bound is sharp: $e^z$ is not constant and omits exactly the value $0$. For polynomials no value is omitted, by the fundamental theorem of algebra.

Going deeper

For a student

The theorem has a version for meromorphic functions: a nonconstant meromorphic function on $\C$ omits at most two values of the Riemann sphere, as $\tan z$ omits $i$ and $-i$. Picard's great theorem is a local version near an essential singularity and implies the little theorem: if $f$ is not a polynomial, then $f(1/z)$ has an essential singularity at $0$. The second main theorem of Nevanlinna theory gives a quantitative form: for three distinct values, the solutions of $f(z) = a$ must be numerous. Through Bloch's principle, the theorem is linked to Montel's theorem on normal families.