1549055259-Ubiquitous_Quasidisk__The__Gehring_

1.3. MODULUS ESTIMATES

Thus p E adm(r),

whence

mod(r):::; r p(z)^2 dm = r p'(f(z))^2 lf'(z)l^2 dm

JR2 JD

= r p'(w)^2 dm:::; r p'(w)^2 dm,

JD, JR

mod(f):::; inf r p'(w)^2 dm = mod(f(f)).

p' JR 2

Now take the infimum over all such p.

Finally we obtain

mod(f) = mod(J(f))

by applying the above argument to f-^1.

7

D
If the curves 'Y E r are disjoint arcs, we may think of them as homogeneous elec-
tric wires. Then the modulus mod(r) is a conformally invariant electrical transcon-
ductance for the family of wires 'Y and the extremal length >.(r) is the total electrical
resistance of the system. In particular, mod(f) is big if the curves 'Y E r are short
and plentiful and small if the curves 'Y are long or scarce.
The following properties show that mod(r) is also an outer measure on the

curve families r in R
2
:
1° mod(0) = 0.
2° mod(ri) :::; mod(f2) if r1 c r2.
3° mod(LJj rj):::; I:j mod(rj).
Finally the conformal invariant mod(r) yields a third characterization for qua-
siconformal mappings.

THEOREM 1.2.2 (Ahlfors [7]). A homeomorphism f: D -t D' is K-quasicon-
f ormal if and only if

1

K mod(r):::; mod(J(f)):::; K mod(f)

for each family r of curves in D.

1.3. Modulus estimates
Estimates for the moduli of various curve families are useful tools for study-
ing geometric properties of conformal and quasiconformal mappings. We derive
here three simple modulus estimates and a distortion theorem for quasiconformal
mappings of the plane which we will need later.

LEMMA 1.3.1. Suppose that R = R(O, a, a+ i, i) is the rectangle with vertices

at 0, a, a+ i, i where a > 0 and suppose that r is the family of curves which join

the horizontal sides of fJR in R. Then

mod(r) =a.

PROOF. The segment 'Y = { z : x + i y : 0 < y < 1} is in r for 0 < x < a. Hence
if p E adm(f), then by the Cauchy-Schwarz inequality,

(

1 ) 2 1

1:::; 1 p(x+iy)dy :::; 1 p(x+iy)^2 dy