1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.12. QUADRILATERALS AND HARMONIC QUASISYMMETRY 57

Next if Q = D(w 1 ,w 2 ,w 3 ,w 4 ) is a quadrilateral in D with mod(Q) = 1, then
mod(f(Q)) = 1 and [z1, z2, z3, z4] = 2 where Zj = f(wj), by Corollary 3.10.3. By
Lemma 3.12.9, inequality (3.12.11) holds with a= a(b). Hence by Lemma 3.10.4

mod(Q*) = mod(B(h(z4), h(z3), h(z2), h(z1))):::; c

where c = c(a) since [h(z1), h(z2), h(z3), h(z4)] = [h(z4), h(z3), h(z2), h(z1)] is bound-
ed. We conclude that D satisfies the conjugate quadrilateral inequality. D