Montel's fundamental normality test
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The everyday Montel theorem buys normality with a numerical bound on the functions. Its far deeper cousin — sometimes called the great Montel theorem — buys the very same compactness from a purely geometric hypothesis with no bound at all: it is enough that every function in the family AVOIDS two fixed values. If none of your holomorphic functions ever equals, say, 0 or 1, the family is automatically normal, no matter how large the functions get. Missing just two targets is, astonishingly, as good as being bounded.
Precisely: fix two distinct complex numbers a and b. Let F be a family of holomorphic functions on a region D such that no f in F ever takes the value a or the value b on D. Then F is normal on D (where the limit infinity is now allowed). The mechanism behind it is the hyperbolic geometry of the thrice-punctured sphere: the plane with the two omitted values (and infinity) removed carries a complete hyperbolic metric, and every function avoiding a and b is distance-decreasing for that metric. That contraction is exactly the equicontinuity needed for an Arzela-Ascoli/Montel argument — boundedness is replaced by metric contraction. The result generalizes and contains Picard's theorems: a nonconstant entire function omitting two values cannot exist, because such a family would be normal in a way that forces the function to be constant.
This test is the engine of complex dynamics, where one studies the iterates f, f∘f, f∘f∘f, ... of a rational or entire map. The Fatou set is by definition where this family of iterates is normal, and the fundamental normality test is the standard tool for proving normality on candidate regions; the chaotic Julia set is the complement. Honest caveat: the number TWO is sharp. Omitting only one value is not enough — e.g. the entire functions e^(nz) all avoid 0 yet do not form a normal family near the imaginary axis. And here the limit of a convergent subsequence may be the constant infinity, so this is normality in the extended (spherical) sense.
Let F be all holomorphic functions on a region D that never take the values 0 or 1. By the fundamental normality test, F is normal — even though the functions in F may be unbounded. By contrast { e^(n z) : n = 1, 2, 3, ... } omits only the single value 0 and is NOT normal near the imaginary axis, where the moduli oscillate wildly.
Omitting two values forces normality; omitting only one (here 0) does not.
The threshold of exactly two omitted values is sharp and is the same threshold behind Picard's theorems. Here a subsequential limit may be the constant infinity, so this is normality in the spherical sense, not the bounded-holomorphic sense.