1549055259-Ubiquitous_Quasidisk__The__Gehring_

1.3. MODULUS ESTIMATES 9 FIGURE 1.3 PROOF. Choose z1 E C1 and z2 E C2 so that iz1 - z2I = dist(C 1 , C2) and set { 1 /a ifzEB(z1 ...

10 1. PRELIMINARIES FIGURE 1.5 THEOREM 1.3.4. If f : R^2 -+ R^2 is K-quasiconformal and if lz2 - zol :<::::: lz1 - zol, then ...

1.3. MODULUS ESTIMATES 11 A more detailed reasoning yields the following sharp estimate for the constant c in (1.3.5), namely c ...

12 1. PRELIMINARIES 1.4. Quasidisks We come now to the principal object of study in this book. DEFINITION 1.4.1. A domain Dis a ...

h 1.4. QUASIDISKS I I I I I FIGURE 1.6 13 Hence if c = 8 eK 2 , then Theorem 1.3.4 implies that for each arc 'Y' C I'' there exi ...

14 1. PRELIMINARIES FIGURE 1.7 for each choice of x in (0, oo). On the other hand, x < If (zi) - f (z2) I :::; c If (z1) -f ( ...

1.4. QUASIDISKS 15 Next set 3 1 1 Z1 = 4' Z2 = 4' Z3 = -4, and l+i -l+i 1-i -1-i W1 = - 2 -, W2 = - 2 -, W3 = - 2 -, W4=- 2 -, a ...

16 1. PRELIMINARIES FIGURE 1.9 -2 -2 EXAMPLE 1.4. 7. For a nonconstant meromorphic function f: R --+ R , the iterates r(z) = f 0 ...

1.5. WHAT IS AHEAD 17 In the remainder of Part 1 (Chapters 2 to 7) we will consider properties in each of these categories. A nu ...

...

CHAPTER 2 Geometric properties We begin our list of characterizations for a quasidisk D with six geometric properties. These inc ...

20 \ \ I I I I ' ' GEOMETRIC PROPERTIES FIGURE 2.1 Let "Y be the open segment (z 1 ,z 2 ) and suppose that "Yn8D = 0. Then "YU ...

2.1. REFLECTION 21 -2 A domain DC R is a Jordan domain if and only if it admits a reflection fin its boundary. What else can we ...

22 2. GEOMETRIC PROPERTIES Example 1.4.5 shows that the boundary fJD of a quasidisk D can be an ex- tremely complicated Jordan c ...

where 2.2. THE THREE-POINT CONDITION a = { c 1 sc(a) if 0 <a:::; 7r/2, if 7r / 2 < a < 7r. 23 The following counterpart ...

24 2. GEOMETRIC PROPERTIES / 1 FIGURE 2.2 THEOREM 2.2.7 (Alestalo-Herron-Luukkainen [10]). If inequality (2.2.3) holds for z1, z ...

2.3. REVERSED TRIANGLE INEQUALITY 25 With (2.2.9) established, it is quite easy to finish the proof in the special case. In fact ...

26 2. GEOMETRIC PROPERTIES If Dis a Jordan domain with oo E 8D, then D satisfies the three-point condition if for some constant ...

2.4. LINEAR LOCAL CONNECTIVITY 27 E er FIGURE 2.3 Thus a Jordan domain D is a quasidisk if and only if it satisfies the reversed ...

28 2. GEOMETRIC PROPERTIES PROOF. Choose z 1 , z 2 EE\ B(z 0 , r) and let 2 z - zo f(z ) = r I 12 + zo. z - zo Then f (z 1 ) , f ...