54.3.1 Montel’s Theorem
Theorem 54.3.1 Let Ω be an open set in ℂ and let ℱ denote a set of analytic
functions mapping Ω to B
⊆ ℂ. Then there exists a sequence of functions
n=1∞ and an analytic function, f such that fn
on every compact subset of
Proof: First note there exists a sequence of compact sets, Kn such that
Kn ⊆ intKn+1 ⊆ Ω for all n where here intK denotes the interior of the set K, the
union of all open sets contained in K and ∪n=1∞Kn = Ω. In fact, you can verify that
Then there exist positive numbers,
such that if z ∈ Kn,
Now denote by ℱn
the set of
restrictions of functions of ℱ
Then let z ∈ Kn
and let γ
+ δneit,t ∈
It follows that for z1 ∈ B
and f ∈ℱ,
It follows that ℱn
is equicontinuous and uniformly bounded so by the Arzela Ascoli
theorem there exists a sequence,
which converges uniformly on Kn.
converge uniformly on K1.
Then use the Arzela Ascoli theorem applied to
this sequence to get a subsequence, denoted by
which also converges
uniformly on K2.
Continue in this way to obtain
uniformly on K1,
Now the sequence
is a subsequence of
and so it converges uniformly on Km
for all m.
short, this is the sequence of functions promised by the theorem. It is clear
converges uniformly on every compact subset of Ω because every such set
is contained in Km
for all m
large enough. Let f
be the point to which
is a continuous function defined on Ω.
Yes it is by Lemma 50.3.13
. Alternatively, you could let T ⊆
Ω be a triangle.
Therefore, by Morera’s theorem, f is analytic.
As for the uniform convergence of the derivatives of f, recall Theorem 50.7.25 about
the existence of a cycle. Let K be a compact subset of int
closed oriented curves contained in
such that ∑
= 1 for every
z ∈ K
. Also let η
denote the distance between
. Then for z ∈ K,
Thus you get uniform
convergence of the derivatives.
Since the family, ℱ satisfies the conclusion of Theorem 54.3.1 it is known as a normal
family of functions. More generally,
Definition 54.3.2 Let ℱ denote a collection of functions which are analytic on
Ω, a region. Then ℱ is normal if every sequence contained in ℱ has a subsequence
which converges uniformly on compact subsets of Ω.
The following result is about a certain class of fractional linear transformations.
Recall Lemma 51.4.7 which is listed here for convenience.
Lemma 54.3.3 For α ∈ B
Then ϕα maps B
one to one and onto B
= ϕ−α, and
The next lemma, known as Schwarz’s lemma is interesting for its own sake but will
also be an important part of the proof of the Riemann mapping theorem. It was stated
and proved earlier but for convenience it is given again here.
Lemma 54.3.4 Suppose F : B
, F is analytic, and F
. Then for
all z ∈ B
If equality holds in 54.3.4 then there exists λ ∈ ℂ with
Proof: First note that by assumption, F
has a removable singularity at 0 if
its value at 0 is defined to be F′
By the maximum modulus theorem, if
< r <
Then letting r → 1,
this shows 54.3.3 and it also verifies 54.3.4 on taking the limit as z → 0. If equality holds
in 54.3.4, then
achieves a maximum at an interior point so
by the maximum modulus theorem. Since F
it follows F′
This proves the lemma.
Definition 54.3.5 A region, Ω has the square root property if whenever f,
ℂ are both analytic ,
it follows there exists ϕ
: Ω → ℂ such that ϕ is analytic and f
The next theorem will turn out to be equivalent to the Riemann mapping