54.5.2 The Little Picard Theorem
Now here is the little Picard theorem. It is easy to prove from the above.
Theorem 54.5.5 If h is an entire function which omits two values then h is a
Proof: Suppose the two values omitted are a and b and that h is not constant. Let
omits the two values 0 and 1.
be defined in
. Then H
is clearly not of the form
because then it would have
values equal to the vertices ln
or else be constant neither of
which happen if h
is not constant. Therefore, by Liouville’s theorem, H′
be unbounded. Pick ξ
is such that H
contains no balls of radius larger than
But by Lemma 54.5.3 H
contain a ball of radius
a contradiction. This proves Picard’s
The following is another formulation of this theorem.
Corollary 54.5.6 If f is a meromophic function defined on ℂ which omits three
distinct values, a,b,c, then f is a constant.
Proof: Let ϕ
the function, h
= ϕ ∘ f.
misses the three points ∞,
meromorphic and does not have ∞
in its values, it must actually be analytic. Thus h
an entire function which misses the two values 0 and 1.
is constant by