1549055259-Ubiquitous_Quasidisk__The__Gehring_

h

1.4. QUASIDISKS

I

I

I

I

I

FIGURE 1.6

13

Hence if c = 8 eK

2

, then Theorem 1.3.4 implies that for each arc 'Y' C I'' there

exists an arc 'YE r such that h('Y) c "(^1 • Thus adm(h(I')) c adm(I''), whence

mod(h(I')) ~ mod(I'')

and

K 2 > mod(h(I')) > 27r-a log(b/a)

• mod(r) - a log(b/a)+2log(c)
  We conclude that
  K2 > 27r - a


• a
  by letting bf a-+ oo.
  Finally reversing the roles of S(a) and S*(a) in the above argument yields
  K2> a


• 27f-O
  and hence (1.4.3).

DEFINITION 1.4.4. A domain Dis a sector of angle a if it is the image of S(a)

under a similarity mapping.

Our second example is a simple Jordan domain that is not a quasidisk.

EXAMPLE 1.4.5. The half-strip

D = { z = x + i y : 0 < x < oo, IYI < 1}

is not a quasidisk.

We shall show that there exists no quasiconformal self-mapping f of R

2
which
maps H onto D. By performing a preliminary Mobius transformation, we need
only consider the case where f(oo) = oo.
Suppose that f is a K-quasiconformal self-mapping of R^2 with f(H) = D, set
w 1 = x + i, w2 = 0, W3 = x - i, and let Zi = 1-^1 (wi) for i = 1, 2, 3. Then z1, z2, z3
is an ordered triple of points on 8H with

lz1 - z2I < lz1 - z3I