Theorem 17.6.1Let Ω be an open set in ℂ and let ℱ denote a set of analytic functions mapping
Ω to B

(0,M )

⊆ ℂ. Then there exists a sequence of functions from ℱ,

{fn}

_{n=1}^{∞}and an analyticfunction f such that for each k ∈ ℕ, f_{n}^{(k)
}converges uniformly to f^{(k)
}on every compact subset of
Ω. Here f^{(k)
}denotes the k^{th}derivative.

Proof: First note there exists a sequence of compact sets, K_{n} such that K_{n}⊆intK_{n+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}^{∞}K_{n} = Ω. In fact, you can verify that B

(0,n)

∩

{z ∈ Ω : dist(z,ΩC ) ≤ 1 }
n

works for K_{n}. Then there
exist positive numbers, δ_{n} such that if z ∈ K_{n}, then B

(z,δn )

⊆intK_{n+1}. Now denote by ℱ_{n} the set of
restrictions of functions of ℱ to K_{n}. Then let z ∈ K_{n} and let γ

(t)

≡ z + δ_{n}e^{it},t ∈

[0,2π]

. It follows that
for z_{1}∈ B

(z,δn)

, and f ∈ℱ,

| ∫ ( ) |
|f (z)− f (z )| = ||-1- f (w) --1--− ---1-- dw||
1 |2πi γ w − z w − z1 |
1 ||∫ z − z1 ||
≤ 2π-|| f (w )(w−-z)(w-−-z-)dw||
γ 1

Letting

|z1 − z|

<

δn
2

,

M |z − z1|
|f (z)− f (z1)| ≤ 2π-2πδn-δ2∕2--
n
≤ 2M |z −-z1|.
δn

It follows that ℱ_{n} is equicontinuous and uniformly bounded so by the Arzela Ascoli theorem there exists a
sequence,

{fnk}

_{k=1}^{∞}⊆ℱ which converges uniformly on K_{n}. Let

{f1k}

_{k=1}^{∞} converge uniformly on K_{1}.
Then use the Arzela Ascoli theorem applied to this sequence to get a subsequence, denoted by

{f2k}

_{k=1}^{∞}
which also converges uniformly on K_{2}. Continue in this way to obtain

{fnk}

_{k=1}^{∞} which converges
uniformly on K_{1},

⋅⋅⋅

,K_{n}. Now the diagonal sequence

{fnn}

_{n=m}^{∞} is a subsequence of

{fmk}

_{k=1}^{∞} and so
it converges uniformly on K_{m} for all m. Denoting f_{nn} by f_{n} for short, this is the sequence of
functions promised by the theorem. It is clear

{fn}

_{n=1}^{∞} converges uniformly on every compact
subset of Ω because every such set is contained in K_{m} for all m large enough. (Why?) Let
f

(z)

be the point to which f_{n}

(z)

converges. Then f is a continuous function defined on Ω.
Is f analytic? Yes it is by Lemma 15.5.5. Alternatively, you could let T ⊆ Ω be a triangle.
Then

∫ ∫
f (z)dz = lnim→∞ fn (z)dz = 0.
∂T ∂T

Therefore, by Morera’s theorem, f is analytic.

As for the uniform convergence of the derivatives of f, recall Theorem 16.5.2 about the existence of a
cycle. Let K be a compact subset of int

(Kn )

and let

{γk}

_{k=1}^{m} be closed oriented curves contained
in

int (K )∖ K
n

such that ∑_{k=1}^{m}n

(γk,z)

= 1 for every z ∈ K. Also let η denote the distance between ∪_{j}γ_{j}^{∗} and
K,

η ≡ inf{ |z − w | : z ∈ K, w ∈ ∪ γ∗}
jj

It follows that η > 0. (Why? In general, two disjoint compact sets are at a positive distance from each
other. Show this is so.) Then for z ∈ K,

| | || m ∫ ||
||f(k)(z)− f(k)(z)|| = ||-k!∑ f (w)−-fn-(w-)dw||
n ||2πij=1 γj (w − z)k+1 ||
m
≤ -k!||f − f || ∑ (length of γ )-1-.
2π k Knj=1 k ηk+1

where here

||fk − f ||

_{Kn}≡ max

{|fk(z) − f (z)| : z ∈ Kn }

. Thus you get uniform convergence of the
derivatives. ■

Another surprising consequence of this theorem is that the property of being one to one is
preserved.

Lemma 17.6.2Suppose h_{n}is one to one, analytic on Ω, and converges uniformly to h on compactsubsets of Ω along with all derivatives. Then h is also one to one.

Proof:Pick z_{1}∈ Ω and suppose z_{2} is another point of Ω. Since the zeros of h − h

(z1)

have no limit
point, there exists a circular contour bounding a circle which has z_{2} on the inside of this circle but not z_{1}
such that γ^{∗} contains no zeros of h − h

(z1)

.

PICT

Using the theorem on counting zeros, Theorem 17.3.1, and the fact that h_{n} is one to one,

. This shows
that h is one to one since z_{2}≠z_{1} was arbitrary. ■

Since the family, ℱ satisfies the conclusion of Theorem 17.6.1 it is known as a normal family of
functions. More generally,

Definition 17.6.3Let ℱ denote a collection of functions which are analytic on Ω, a region. Then ℱis normal if every sequence contained in ℱ has a subsequence which converges uniformly on compactsubsets of Ω.