1549055259-Ubiquitous_Quasidisk__The__Gehring_

120 9. SECOND SERIES OF IMPLICATIONS

and

'V;'(s) - "2'1;(s)^1 2 < I ( y(z(s)) f" ) I z'(s) I -^1 "2 I f" y(z(s))^12

I

(

::::; y(z(s)) f" ) I - "2 1 ( f" y(z(s)) )^21 = IS1(z(s))I

c

<------

• (length(!) - s)2
  a.e. in [a, b). Thus

'VJ'(s) - ¢'(s) < ~('V;(s)^2 - ¢(s)^2 ) = ('V;(s) - ¢(s)) ( 'V;(s); ¢(s))

a.e. in [a, b), and we can apply Lemma 9.1.4 with u = '1; - ¢and v = ~('VJ+¢) to

conclude that

(9.1.8)

I

r f'(z(s)) I < 'V;(s)::::; ¢(s) = min(s,length(I)) ~

for s E [a,b).

Finally, we claim that b = length(!). Otherwise, (9.1.8) would imply that

f" / f' is bounded near z(b) and hence that f is analytic at z(b). In particular, we

would then get b' E (b, length(!)) such that f is analytic at z(s) for s E [a, b'),

contradicting the way b was chosen. This completes the proof of (9.1.7) and hence

of Lemma 9.1.5. 0

We show now that if D is uniform, then cr(D) > 0, i.e., functions analytic in D

are injective if their Schwarzian derivatives are small compared with the square of

the hyperbolic metric in D.

THEOREM 9.1.9. Suppose D is a simply connected domain in R^2 and suppose
there exists a constant c > 1 such that each z 1 , z 2 E D can be joined by an arc
"f c D so that

(9.1.10)

(9.1.11)

length(!) ::::; c iz1 - z2i,

min length(lj) ::::; c d1st(z, 8D)

J=l,2

for each z E "(, where "(1, "(2 are the components of "( \ { z}. If f is meromorphic

and locally injective in D and if

2 1

(9.1.12) sup IS1(z)IPD(z)- < - 3 ,

zED 16 C

then f is injective in D. Hence, in particular,

1

(9.1.13) cr(D) 2

16

c 3 ·

PROOF. Choose z1, z2 E D and let"( be an arc joining these points for which

(9.1.10) and (9.1.11) hold. Then by (3.2.1) and (9.1.13),

I

1 2 1 1

S1 (z) I ::::; 16 c3 PD(z) ::::; 4 c3 dist(z, 8D) 2

1 1

< - ----,--------.,---,-------,-

• 4c min(s,length(I)-s)2

(9.1.14)