The univalent Bloch-Landau constant, harmonic symmetry and ...

Next we write Γ 1 for the admissible family of arcs formed by the
straight line segment ˜γ 1 = [r,1], together with is rotations ˜γ k+1 =
e 2πki/n˜γ 1 , k = 1,2, ..., n−1,whererischosensothatω ( 0,˜γ 1 ;D(Γ 1 ) ) =
1/(2n). We write g for the conformal map of D onto D(Γ 1 ) for which
g(0) = 0 and g(Ĩk) = ˜γ k , for 1 ≤ k ≤ n. Then
S = g ◦T
is a quasiconformal map of the unit disk onto D(Γ 1 ). The map S
extends continuously to the boundary of D. Tracing the boundary
correspondence under the mappings T, and then g, shows that it is a
quasiconformal glueing of the intervals I k , in that
S(ζ) = S ( φ k (ζ) ) , ζ ∈ I k , 1 ≤ k ≤ n.
By the Measurable Riemann Mapping Theorem, we can now correct S
to a conformal glueing by making a quasiconformal map R of D onto
D for which R(0) = 0 and
f = R◦S
is conformal. Then (4.1) holds. The arcs we are looking for are then
γ k = R(˜γ k ), for 1 ≤ k ≤ n and Γ = {γ 1 ,γ 2 ,...,γ n }. Since f is
conformal and f(0) = 0, it preserves harmonic measure at 0. Thus
ω ( 0,γ k ;D(Γ) ) = ω ( 0,f −1 (γ k );D ) = ω ( 0,I k ;D) = a k .
This is (T2.1). Moreover, since f is a conformal glueing on the interval
I k , the domain D(Γ) is harmonically symmetric in γ k with respect to
0, which is (T2.2). Finally, f maps each arc J k on the unit circle onto
the anti-clockwise arc of the unit circle joining the endpoints ζ k of γ k
and ζ k+1 of γ k+1 on the unit circle, so that (T2.3) follows. The fact
that the arcs are real analytic comes from the control we have on the
quasiconformality of the maps constructed.
This construction depends continuously on the parameters a k and
b k . In the limit as one or more of the parameters b k approach zero, it
leads to a configuration satisfying (T2.1), (T2.2) and (T2.3) in which
two or more of the curves have a common endpoint on the unit circle.
This covers the proof of existence in all cases in Theorem 2.
We now deal with the uniqueness of the configuration we have just
now constructed. We suppose that Γ 1 = {γ1,γ 1 2,...,γ 1 n} 1 and Γ 2 =
{γ1,γ 2 2,...,γ 2 n} 2 are sets of admissible arcs that satisfy (T2.1), (T2.2)
and (T2.3) (with Γ replaced by Γ 1 and by Γ 2 as necessary). We assume
thatthearcsinΓ 1 arerealanalytic, butthoseinΓ 2 neednotbe. Having
set up the intervals I 1 to I n and J 1 to J n as in the proof of existence,
we consider conformal mappings f 1 : D → D(Γ 1 ) and f 2 : D → D(Γ 2 )
with f 1 (0) = 0, f 2 (0) = 0 and with f 1 (I k ) = γk 1, f 2(I k ) = γk 2, for
each k between 1 and n. Then f = f 2 ◦f1 −1 is a conformal map from
D(Γ 1 ) to D(Γ 2 ). Moreover, f extends continuously from D(Γ 1 ) to D
because of the harmonic symmetry. By the regularity of the arcs in Γ 1 ,
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