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

We consider an (i+1)-face f 1 of Σ 1 , the i-faces that bound it, and the
corresponding (i + 1)-face f 2 of Σ 2 . Assuming that T maps each of
these i-faces onto the corresponding i-face of Σ 2 , it follows fromthe key
property of T that the image of the i-faces bounding f 1 has winding
number 1(mod2) about each interior point of f 2 . Consequently, T is
onto f 2 .
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4.3. Harmonically symmetric arcs don’t grow. It is natural to
ask whether the harmonically symmetric arcs γ k in Theorem 3 can be
described by means of a differential equation of Löwner type. As our
final result, we show that this is not possible, even in the case of two
harmonically symmetric arcs. We note that if the data in Theorem 3
is symmetric with respect to R, then, by construction, the resulting
harmonically symmetric curves can be taken symmetric with respect
to R.
Theorem 4. We suppose that ζ lies on the upper half of the unit circle
and that a 1 and a 2 lie in (0,1) with a 1 < a 2 . Theorem 3 is applied twice
to construct two pairs of harmonically symmetric curves, {γ 1 ,γ 1 } and
{γ 2 ,γ 2 } respectively, the first from the data a 1 , a 1 , ζ, ζ, and the second
from the data a 2 , a 2 , ζ, ζ. Then γ 1 ⊈ γ 2 except in the case when ζ = i.
Proof. Let us suppose that γ 1 ⊆ γ 2 . Since D\{γ 1 ∪¯γ 1 } is harmonically
symmetric in ¯γ 1 with respect to 0, ¯γ 1 is a geodesic arc in D\γ 1 . Since
D\{γ 2 ∪¯γ 2 } is harmonically symmetric in ¯γ 2 with respect to 0, ¯γ 2 , and
hence its subarc ¯γ 1 , is a geodesic arc in D \γ 2 . Thus ¯γ 1 is a geodesic
arc with respect to both the domain D\γ 1 and the domain D\γ 2 .
We write Γ 1 for the full geodesic in D\γ 1 of which ¯γ 1 is a part and
write Γ 2 for the full geodesic in D\γ 2 of which ˜γ 1 is a part. Both Γ 1
and Γ 2 pass through the origin. We map D\γ 1 onto the unit disk by
a conformal map f 1 so that Γ 1 is mapped onto (−1,1) and map D\γ 2
onto the unit disk by a conformal map f 2 so that Γ 2 is mapped onto
(−1,1).
We consider the map g = f 1 ◦ f2 −1 , which maps the unit disk into
itself conformally. Moreover, g is real-valued on f 2 (¯γ 1 ), which itself is
a subinterval of (−1,1). Thus g is real-valued on the entire interval
(−1,1). Since f2 −1 (−1,1) = Γ 2 , it then follows that
f 1 (Γ 2 ) ⊆ (−1,1) = f 1 (Γ 1 ),
so that
Γ 2 ⊆ Γ 1 .
There arenow two possible geometric situations to consider, depending
on where the geodesic Γ 2 might end (both it and Γ 1 begin at the
endpoint of ¯γ 1 on the unit circle). Suppose that Γ 2 were to end at a
boundary point of D \ γ 1 . In this case, Γ 2 and Γ 1 coincide and the
final step is to recall that the harmonic measure of the boundary is
split evenly in two along a hyperbolic geodesic. Let E be that part of
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