1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.10. QUADRILATERALS 49 To show that these bounds are sharp, suppose that 0 < a :::; 7r and for 1 < r < oo let Q = D(z1 ...

50 3. CONFORMAL INVARIANTS r FIGURE 3.3 x = m 2-n where n = 0, 1, 2, .... Hence ¢(x) = x for all x E R^1 by continuity and h(z) ...

3.11. EXTREMAL DISTANCE PROPERTY 51 for each pair of continua C 1 and C 2 in D. The bound in (3.11.2) is sharp. Example 3.11.1 s ...

52 3. CONFORMAL INVARIANTS Example 3.11.8 is a reformulation of the level-set inequality, first established by Hayman and Wu [79 ...

3.12. QUADRILATERALS AND HARMONIC QUASISYMMETRY Now suppose that Z1' Z2, Z3 is a triple of points in aB with lz1 - z2I = lz2 - z ...

54 3. CONFORMAL INVARIANTS The square of the length l of the diagonal (z 1 , z3) is then given by l^2 = c^2 + d^2 - 2cdcose = a^ ...

3.12. QUADRILATERALS AND HARMONIC QUASISYMMETRY 55 by (3.12.7), and by interchanging the pairs of indices (1, 2) and (3, 4) and ...

56 3. CONFORMAL INVARIANTS f h g FIGURE 3.4 whenever lz 1 - z2I = lz2 - z3I, i.e., whenever z1,z2 and z2,z3 are the endpoints of ...

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, t ...

...

CHAPTER 4 Injectivity criteria Suppose that 1 is a mapping which is locally injective in a domain D. When can we conclude that 1 ...

60 4. INJECTIVITY CRITERIA The following shows that the size of the Schwarzian relative to the hyperbolic metric is related to t ...

4.1. MEROMORPHIC FUNCTIONS 61 The conclusion in Theorem 4.1.7 can be strengthened when Dis a disk or half- plane. In this case f ...

62 4. IN JECTIVITY CRITERIA THEOREM 4.1.11 (Osgood [141]). If f is analytic and injective in a simply connected domain D C R^2 , ...

4.2. LOCALLY BILIPSCHITZ MAPPINGS 63 EXAMPLE 4.2.2. If f is locally 1-bilipschitz in a domain D c R^2 , then f is injective in D ...

64 4. INJECTIVITY CRITERIA SKETCH OF PROOF. Let D' = f(D) and suppose that g is locally L'-bilipschitz in D' with L' < L(D) L ...

4.3. LOCALLY QUASICONFORMAL MAPPINGS 65 for locally quasiconformal mappings. It turns out t hat the BMO (bounded mean oscillatio ...

66 4. INJECTIVITY CRITERIA Then f is analytic in B(O, r) with u =Re(!) and differentiation yields j'(O) = .!_ (2n u(reie) de=.!_ ...

4.4. JACOBIAN OF A CONFORMAL MAPPING 67 The following quasiconformal analogue of Theorems 4.1.4 and 4.2.3 shows that the implica ...

68 4. INJECTIVITY CRITERIA in D. Hence h is constant in D, g = af + b where a and b are constants, and g is injective in D. Thus ...