Math 213a: Complex analysis
Problem Set #8 (12 November 2003):
Harmonic functions and their uses, cont’d
First, an observation on the coefficients of the linear equations used to determine
the logarithm of our conformal map of a finitely connected region (Ahlfors,
V.3.1, Ex.2 (p.204)):
1. Let Ω ⊂Cbe a region whose boundary consists of the simple closed curves
C1, . . . , Cn. As in Ahlfors V.3.1, for j, k ∈ {1, . . . , n}let αjk =HCj∗dωk,
where ωkis the harmonic measure of Ckwith respect to Ω, and Cjis
traversed in the positive direction relative to Ω. Prove that αjk =αkj .
Next, some problems on the nice special case of a doubly-connected region.
We begin with an elementary problem (that is, a problem not requiring the
machinery of (sub)harmonic functions from Chapter V) that picks up a thread
from the fourth problem set, where we studied the action of PGL2(C) on pairs
of disjoint circles.
2. If Ω is the doubly connected region bounded by two disjoint circles, show
that the conformal bijections of Ω to an annulus are fractional linear
transformations. Deduce that the subgroup of PGL2(C) that preserves
two disjoint circles is conjugate in PGL2(C) to the group of fractional
linear transformations of the form z7→ cz for some c∈C∗with |c|= 1.
Obtain a similar description of the stabilizer in PGL2(C) of two circles
that are tangent or meet in two points.
3. Let Ω be the annulus {z∈C:r < |z|< R}. Prove that if uis a bounded
harmonic function on Ω then, for any ρ∈(r, R),
log R
rI|z|=ρ
∗du ≤2πsup
z
u(z)−inf
zu(z),
with equality if and only if there exist C0, C1∈R, with C1≥0, such that
u(z) = C0+C1log |z|for all z∈Ω. Conclude that the only conformal
bijections of Ω are z7→ cz and z7→ cRr/z (|c|= 1).
4. Suppose Ω and Ω0∈Care doubly connected regions conformally equivalent
to the annuli r < |z|< R and r0<|z|< R0. Let their complements in
P1(C) have components E1, E2and E0
1, E0
2, with E1, E0
1containing ∞.
Prove that if Ej⊇E0
jfor j= 1,2 (whence Ω ⊆Ω0) then R/r ≤R0/r0,
with equality only when Ω = Ω0.
5. Suppose Ω is a simply-connected bounded subset of C, and fix z0∈Ω. For
each r > 0 such that Ω contains the closed disc {z:|z−z0| ≤ r}, the region
Ωr:= {z∈Ω : |z−z0|> r}is doubly connected, and thus conformal with
an annulus Arwith outer radius 1. Let ρ(r) be the inner radius of Ar. By
