1549055259-Ubiquitous_Quasidisk__The__Gehring_

66 4. INJECTIVITY CRITERIA

Then f is analytic in B(O, r) with u =Re(!) and differentiation yields

j'(O) = .!_ (2n u(reie) de=.!_ (2n u(reie) ~ u(O) de.

7r lo re•^8 7r lo re•^8

Since !grad u(O)I = IJ'(O)I, we have

lgrad u(O)I::::; .!_ f

2

1r r-^1 lu(rei^8 ) - u(O)I de.

7r lo

From this together with inequality (3.2.1)

lgrad u(O)I PD(o)-^1 ::::; lgrad u(O)l 2d

::::; m(~o) fod !grad u(O)l 7rr^2 dr

::::; m(~o) l

0

lu(z) - u(O)I dm.

Next for (4.3.5) let 'Y be the hyperbolic geodesic joining 0 to z in D. Then

lu(z) - u(O)I ::::; c 1 poi dz I = c hD(z, 0),

where

and

c =sup lgrad u(z)I PD(z)-^1

zED

m(~o) l

0

lu(z) - u(O)ldm::::; m(~o) lo hD(z, 0) dm::::; 2c

by Lemma 4.3.2. D

COROLLARY 4.3.6 (Astala-Gehring [14]). If f is analytic with J' =/= 0 in D c

R^2 , then

1 1

• II log J1llBMO(D) :S sup IT1(z)I PD(z)- :S 3 II log J1llBMO(D)·
  4 zED
  PROOF. Let u(z) =log lf'(z)I. Then u is harmonic in D,
  lf"(z)I
  log JJ(z) = 2 log lf'(z)I = 2u(z), ITt(z)I = lf'(z)I =!grad u(z)I,

and the desired conclusion follows from Lemma 4.3.3. D

Corollary 4.3.6 shows that the BMO-norm of log J 1 is a natural alternative

for the pre-Schwarzian derivative Tt when considering injectivity results for locally

conformal mappings. Moreover the following quasiconformal counterpart of The-

orem 4.1.3 and Theorem 4.1.11 suggests that this norm offers a way to extend

results on the injectivity of analytic functions to the class of locally quasiconformal

mappings.

THEOREM 4.3.7 (Reimann [146]). If f is K-quasiconformal in D with J(D) C

R^2 , then

II log JtllBMO(D) ::::; m

where m = m(K) < oo. Moreover, when D = R^2 ,

lim m(K) = 0.

K---71