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24 2. GEOMETRIC PROPERTIES

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FIGURE 2.2

THEOREM 2.2.7 (Alestalo-Herron-Luukkainen [10]). If inequality (2.2.3) holds

for z1, z2 E 8D such that

(2.2.8)

then it holds for all z 1 , z 2 E 8 D with the constant a replaced by 2a.

PROOF. As before we let 11 ,1 2 denote the components of 8D \ {z 1 ,z 2 } and

assume that diam(l 1 ) ::::; diam(l 2 ). More generally, for u, v E 8D n [z 1 , z 2 ] with

z 1 ::::; u < v ::::; z2 we let 11 ( u, v) and 12 ( u, v) denote the comp onents of 8D \ { tl, v}

labeled so that diam(l 1 ( u, v)) ::::; diam(l2 ( u, v)).

Before we discuss the general case we consider the special case where ry 1 n

[z 1 , z 2 ] = { z 1 , z 2 }. See Figure 2.2. Let [uj, Vj] denote the closures of the components

of [z 1 , z2] \ ry 2. Now 8D n [uj, Vj] = { Uj, Vj} and (2.2.3) follows immediately if

11 ( Uj, Vj) :::i 11 for some j. Hence we assume 11 C 12 ( Uj, Vj) for all j. In this case

(2.2.9) 12 c [z1,z2] u LJ11(uj,vj)·

j

To see this, fix z E 12 \ [z 1 , z2] and let w 1 and w 2 be the first points of 12 n [z 1 , z 2 ]

as 12 is traversed from z towards z 1 and z 2 , respectively. We next consider the

subarcs a and f3 say, of 12 joining z 1 , w 1 and w 2 , z 2 , respectively. Then

'7 2 n [z1, z2] =(au i3) n [z1, z2].

Next se lect u E an [z1,z2] and v E j3n [z1,z2] so that

lu - vi= dist(a n [z1, z2], i3 n [z1, z2]).

Then the segment joining u and v (or v and u) equals [uj, Vj J for a suitable j. Since

11 c 12 ( Uj, Vj), the ordering of the points implies t hat z E 11 ( Uj, Vj).