Math 411: Honours Complex Variables - University of Alberta

114 CHAPTER 17. THE RIEMANN MAPPING THEOREM
Let z0 ∈ D1 and let w0 := f(z0). We see that f ′ is differentiable at z0 only if
f ′ (z0) �= 0: for w ∈ D2 \{w0},
f −1 (w)−f −1 (w0)
w −w0
=
f −1 (w)−f −1 (w0)
f(f −1 (w))−f(f −1 (w0)) −→
w→w0
1
f ′ (z0) .
Since f is not constant, f ′ is not identically zero and thus Z(f ′ ) is discrete. We
claim that f(Z(f ′ )) is also discrete. Assume that f(Z(f ′ )) is not discrete. Then there
exist w0 ∈ D2 and a sequence (zn) ∞ n=1 in Z(f′ ) such that w0 �= f(zn) for n ∈ N, but
w0 = lim f(zn). By the bijectivity and continuity of f
n→∞ −1 , we have f−1 (w0) �= zn for
n ∈ N and f −1 (w0) = lim zn. Hence, f
n→∞ −1 (w0) is a cluster point of Z(f ′ ), which is
impossible.
Hence, f−1 is holomorphic on D2\f(Z(f ′ )). Since f−1 is continuous and f(Z(f ′ ))
is discrete, Riemann’s Removability Criterion then yields the holomorphy of f−1 on
all of D2. Thus Z(f ′ ) = ∅.
Corollary 17.2.1. Let D ⊂ C be open and connected, and let f : D → C be
holomorphic and injective. Then Z(f ′ ) = ∅.
Proof. If f is injective, it is not constant. By the Open Mapping Theorem, f(D) is
thereforeopenandconnected. ApplyTheorem17.2withD1 = D andD2 = f(D).
Theorem 17.3 (RiemannMapping Theorem). Let D � C be open and connected and
admit holomorphic square roots, and let z0 ∈ D. Then there is a unique biholomorphic
function f: D → D with f(z0) = 0 and f ′ (z0) > 0.
Proof. Uniqueness: Let g: D → D be another such function. Then f ◦g −1 : D → D is
biholomorphic with (f ◦g −1 )(0) = f(z0) = 0. By Corollary 7.4.1, there exists c ∈ C
with |c|= 1 such that
f(g −1 (z)) = cz
for z ∈ D and thus
f(z) = f(g −1 (g(z))) = cg(z)
for z ∈ D. Differentiation yields f ′ (z) = cg ′ (z) for z ∈ D. Since |c|= 1 and both
f ′ (z0) and g ′ (z0) are real and positive, we conclude that c = 1.
Existence: Let
F := {f: D → D : f is injective and holomorphic with f(z0) = 0 and f ′ (z0) > 0}.
Claim 1. F �= ∅.
Since D �= C, there exists w ∈ C\D. Since D admits holomorphic square roots,
there is a holomorphic function g: D → C such that [g(z)] 2 = z−w for z ∈ D. Note
for z1,z2 ∈ D that g(z1) = ±g(z2) ⇒ z1 = z2. In particular, this means that g is
injective and thus not constant.