1549055259-Ubiquitous_Quasidisk__The__Gehring_

1.3. MODULUS ESTIMATES 9

FIGURE 1.3

PROOF. Choose z1 E C1 and z2 E C2 so that iz1 - z2I = dist(C 1 , C2) and set

{

1 /a ifzEB(z1,a+b),

p(z) =

0 otherwise.

Then each 'YE r either joins C 1 to C 2 in B(z 1 , a+b) or joins 8B(z 1 , b) to 8B(z 1 , a+
b). In either case 'Y contains a subarc of length at least a which lies in B(z 1 , a+ b).
Thus p E adm(r) and

D
We now apply the modulus estimate established above in Lemma 1.3.2 to prove
an elementary distortion theorem for quasiconformal mappings which we will need
in what follows.

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I

I I

'

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FIGURE 1.4

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