1549055259-Ubiquitous_Quasidisk__The__Gehring_

122 9. SECOND SERIES OF IMPLICATIONS

For the general case set g(z) = f(<P(z))<f/(z) where ¢ : B-+D is conformal.

Then g is analytic in B,

lg(z)I ~ CPD(¢(z))l<P'(z)I = CPB(z)

for z EB, and

IJ'(<P(z)) ¢'(z)^2 + f(<P(z))<P"(z)I = lg'(z)I ~ 2cpB(z)^2

= 2cpD(¢(z))^2 l¢'(z)l2

by what was proved above. Hence

lf'(¢(z))I ~ lf(¢(z))l I::~~;; I+ 2cpD(¢(z))

2

~ cpD(¢(z)) I::~~;; I+ 2cpD(¢(z))

2

~ 5cpD(¢(z))

2

,

since from a well-known distortion theorem for conformal maps of B we obtain

I

<P"(z) _ 2lzl

2

I < 4lzl

z <P'(z) 1 - lzl^2 - 1 - lzl^2 '

whence

I

<P"(z) I 2lzl + 4 ,

<P'(z) ~ l - lzl 2 ~ 3pB(z) = 3pD(¢(z))l<P (z)I.

See, for example, Duren [33], Hayman [78]. D

COROLLARY 9.2.4. Suppose that f is analytic and locally injective in a simply

connected domain D C R^2. If.

then

sup ISJ(z)I PD(z)-^2 ~ 5c + ~ c^2.

zED

PROOF. By hypothesis, TJ is analytic with

in D. Hence

ISJ(z)I PD(z)-^2 ~ ITj(z)IPD(z)-^2 + ~ (ITJ(z)I PD(z)-^1 )

2

~ 5c + ~c^2

by Theorem 9.2.1.

The following result is an immediate consequence of Corollary 9.2.4.

COROLLARY 9.2.5. If DC R^2 is simply connected with cr(D) > 0, then

r(D) ~ J25 + cr(D) - 5 > 0.

D