1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.2. SCHWARZIAN AND PRE-SCHWARZIAN DERIVATIVES

for z E "(, where s is the arclength of 'Y measured from z 1 to z.
Let z 0 be the midpoint of"(. If f"(z 0 ) = 0, then

I

f (z^1 ),"( ~(z^2 ) - (z1 - z2) I :::; -

1

• length('Y)
  zo 2c - 1
  c

:::; --Jz1 -z2J < Jz1 -z2J

2c-1

121

by (9.1.14) and Lemma 9.1.5 with 1/4c in place of c and by (9.1.10). Hence f(z 1 ) i-
f (z2).
If f"(z 0 ) ¥ 0, we can choose a Mobius transformation g so that h"(z 0 ) = 0
where h = go f. Then Sh = St and we can apply the above argument to h to
conclude that h(z 1 ) 'I-h(z2), whence f(z 1 ) 'I- f(z2). Thus f is injective in D. D

9.2. Schwarzian and pre-Schwarzian derivatives
Suppose that Dis a simply connected domain and that f is any function which
is analytic and locally injective in D. We show here that if there is a constant a > 0
so that f is injective whenever

JS1(z)J:::; apD(z)^2

in D, then there is a second constant b > 0 so that f is injective whenever

in D. This fact is an easy consquence of the following result. Cf. Duren-Shapiro-
Shields [35], Wirths [167].

THEOREM 9.2.1. Suppose that f is analytic in a simply connected domain D c
R^2. If

(9.2.2) sup Jf(z)J PD(z)-^1 :::; c,
D

then

(9.2.3) sup Jf'(z) J PD(z)-^2 :::; 5c.
D
PROOF. Suppose first that D = B, fix z E B , and choose r > 0 so that
2r^2 = 1 + Jz J^2 , whence

Then
2
lf(()J :::; c 1 - J(J2

for ( E B and with the Cauchy and Poisson integral formulas we obtain

lf'(z)J = I~ r f(() 2 d(I

27ri }l(l=r (( - z)

<-- -. dB

2c r 1 12 71' r^2 - JzJ^2

• 1 - r^2 r^2 - Jz J^2 271' 0 Jr e' e - z J^2
  2c r )2
  1-r 2 2 r - J z^12 < 2cpB(z.