1549055259-Ubiquitous_Quasidisk__The__Gehring_

8.6. QUADRILATERAL INEQUALITY AND QUASIDISKS 109 PROOF. Set f(z) = f(z) for z E H*. Then f is a self-homeomorphism of R^2 which ...

110 8. FIRST SERIES OF IMPLICATIONS for z E H U H * where K = 2k(k + 1). See, for example, Beurling-Ahlfors [23], Lehto-Virtanen ...

8.6. QUADRILATERAL INEQUALITY AND QUASIDISKS 111 and thus (8.6.11) mod(Qr)='!:_μ( 1 ) · 7r Jl + l /u Let Q = g(Q1) and Q* = g*(Q ...

112 8. FIRST SERIES OF IMPLICATIONS 8.7. Reflections and quasidisks Theorem 8 .6. 7 completes a circle of implications to show t ...

8.7. REFLECTIONS AND QUASIDISKS 113 The following result yields a Euclidean analo gue of Theorem 2.1.11. -2 -2 LEMMA 8.7.3. Supp ...

114 8. FIRST SERIES OF IMPLICATIONS for j = 1, 2, ... , n. If, in particular, h('Y) is the segment joining h(zi) and h(z2), then ...

8.8. QUASIDISKS AND DECOMPOSABILITY hyperbolic segment/ joining z 1 and z 2 in D' CD such that for each z E /, length(!) :::; c ...

...

CHAPTER 9 Second series of implications In the last chapter we proved seven ways of characterizing a quasidisk by means of a cir ...

118 9. SECOND SERIES OF IMPLICATIONS 9 .1. Uniform domains and Schwarzian derivatives The arguments in this section are based in ...

9.1. UNIFORM DOMAINS AND SCHWARZIAN DERIVATIVES 119 for z E 'Y between z1 and zo. This inequality then implies that f is bounded ...

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 ( ::::; ...

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 ...

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 ...

9.3. PRE-SCHWARZIAN DERIVATIVES AND LOCAL CONNECTIVITY 123 9.3. Pre-Schwarzian derivatives and local connectivity We saw in the ...

124 9. SECOND SERIES OF IMPLICATIONS PROOF. By Lemma 9.3.l there exist points z 1 , z 2 ED and wo E R^2 \ D such that (9.3.6) . ...

9.3. PRE-SCHWARZIAN DERIVATIVES AND LOCAL CONNECTIVITY 125 standard distortion theorems applied to h(z)/h'(O) imply that 1 -- I ...

126 9. SECOND SERIES OF IMPLICATIONS Then f is analytic in D with f' f:. 0 and I (^1) " I I g( z) g' ( z) I f =a lwl -z-+ 1 +aw ...

9.4. UNIFORM DOMAINS ARE RIGID 127 9.4. Uniform domains are rigid Suppose that D C R^2 is a simply connected domain. In the last ...

128 9. SECOND SERIES OF IMPLICATIONS and by (9.4.4) l(FT F - I)zl^2 = l(IJ(l)l^2 - 1) x + Re(f(l)f(i)) Yl^2 + IRe(f(l)f(i)) xl^2 ...