1549055259-Ubiquitous_Quasidisk__The__Gehring_

118 9. SECOND SERIES OF IMPLICATIONS

9 .1. Uniform domains and Schwarzian derivatives
The arguments in this section are based in large part on the ideas of Martio
and Sarvas in [123]. We begin wit h a lemma which compares the size of the pre-
Schwarzian derivative TJ = f" / f' with the values f assumes at two different points.

LEMMA 9.1.l. Suppose that z 1 ,z 2 ED C R^2 , that/ is a rectifiable open arc
joining z 1 , z 2 in D with midpoint z 0 , and that 0 < c < l. If f is meromorphic and
locally injective in D and if

(9.1.2)

I

f"(z) I c

f'(z ) ::::; min(s,length('Y) - s)

for z E /, where s is the arclength of/ from z 1 to z, then

I

f(ziJ ( ~(zz) - (z1 -z2)I::::; _ c _ length('Y).

I Z Q 1 - C

PROOF. By the triangle inequality it is sufficient to prove that

(9.1.3)

I

f(zj) - f(zo) - (z - zo)I::::; ~ _ c _ length('Y)

f'(zo)^1 2 1 - c

for j = 1, 2. By symmetry we need only consider the case where j = l.
Now (9.1.2) implies that f" / f' and hence f are finite at each z E 'Y· For z E 'Y
let
rr
g(z) = Jzo f (() d(,

where the integral is taken along f. Then

and

eg(z) = f'(z )

f'(zo)

f(z) - f(zo) - (z - zo) = r (eg(() - 1) d(
f'(zo) Jzo
for z E f. If z E / is between z 1 and z 0 , then

I

f" f(z ) I ::::; c ~'

by (9.1.2), whence

lg(z)i::::; }r zo If" f (() I id(I::::; las -;: c d<J =clog(-.;), a

where for convenience of notation we let a = length( 'Y) /2. Therefore

leg(z) -11::::; elg(z)I -1::::; (~r -1

and we obtain

I

f(z) ,-f(zo) - (z - zo)I ::::; r leg(() - 11 Jd(J

f (zo) Jzo

::::; 1a (( ~ r -1) d<J

1 c

=

2 1

_ c length('Y)