1549055259-Ubiquitous_Quasidisk__The__Gehring_

2.4. LINEAR LOCAL CONNECTIVITY 29

c v u

FIGURE 2.4

By performing a final pair of similarity mappings we may assume that L' as
well as L is the real axis and that

J(l) = 1, f(-1) = -1, f(c) = u > 1, f(-c) = v

and it is enough to consider the cases where f (-c) = v where v :::; -u or v 2 u + 2
including v = oo. Then J(E 1 ) and J(E2) will be separated by the annulus { w :
1 < lw I < u}. Finally, using the fact that cross ratios are invariant under Mo bi us
transformations we h ave that

(c1;2 + c-1/2)2 = 4 (c + 1)(1 + c) = 4 (u + 1)(1-v) = h(u v)

(c+c)(l+l) (u-v)(l+l) '

where

Since

1-UV

h(u, v) = 2-- + 2.

u-v

EJh(u,v) _

2

l-u^2 < O

av - (u-v)^2 )

h(u, v) is decreasing in v , and we have that

(c^112 + c-^112 )^2 :::; max h(u, v) = lim h(u, v) = 2(u + 1) < (u + 1)^2

v:S-u v--4--oo

if v :::; -u and

(c^112 + c-^112 )^2 :::; max h(u, v) = lim h(u, v) = (u + 1)^2

v;:::u+2 v-4u+2

if v 2 u + 2. In either case,
g(c) = c1;2 + c-1/2 - 1:::; u

and hence f(Ei) and J(E 2 ) are separated by {w: 1 < lwl < g( c)}.
The extremal situation occurs when
f(l) = 1, f(-1) = -1, f(c) = g(c), f(-c) = g(c) + 2,

in which case

J(C1) = {z: lzl = 1}, J(C 2 ) = {z: lz - (g(c) + 1)1 = l}.

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