1549055259-Ubiquitous_Quasidisk__The__Gehring_

8 1. PRELIMINARIES

i a+i

R

0 a

FIGURE 1.

for 0 < x < a. Thus

L 2 p(z)

2

dm 21a (

1

p(x + iy)

2

dy) dx 2 a

and

Next the function

is in adm(r) and

mod(r) =inf r p(z)^2 dm 2 a.

P }R 2

{

1 if z ER,

p(z) =

0 otherwise

r p(z)^2 dm =a,
}R 2
completing the proof for Lemma 1.3.1. D

LEMMA 1.3.2. If r is a family of curves and if for each t with a < t < b the

circle { z : I z I = t} contains a curve / E r , then

1 b

mod(I') 2 - log -.

27r a

PROOF. See Figure 1.3. If p E adm(r), then

1 :.::: (1 p(z) ldzlr :.::: (

2

7' p(t ei^6 ) t de)

2

:.::: 27rt 1

2

7' p(t ei^6 )^2 t de,

whence

-2^1 log~= {b_2l dt::; fb(f

2

7rp(tei^6 )^2 tde)dt::; r p(z)^2 dm.

7l" a J a 'lt"t J a JO JR 2

Now take the infimum over all such p. D

LEMMA 1.3.3. If r is a family of curves which join continua C 1 and C 2 where

diam( C 1 ) :.::; b,

then