1549055259-Ubiquitous_Quasidisk__The__Gehring_

D. CHAPTER 11 Fourth series of implications In this chapter we prove the following statements for a simply connected domain 1° A ...

150 11. FOURTH SERIES OF IMPLICATIONS 11.1. Harmonic bending and quasidisks We show first that a Jordan domain DC R^2 is a quasi ...

11.1. HARMONIC BENDING AND QUASIDISKS 151 D /z2 ___l__ ...-; -...... ...... / Z1 ---_.. / -_.. ------~·-_.,_ ---··----- lg .1 in ...

152 11. FOURTH SERIES OF IMPLICATIONS -2 G (= R \ 1) s ... ... ..... ---. z n* FIGURE 11.3 map h of R 2 \ f ( /) onto H. Without ...

11.3. HOMOGENEITY AND QUASIDISKS since by monotonicity But by (11.1.6) and so that This means that w(z, f)D,* n &G; G) 2: w( ...

154 11. FOURTH SERIES OF IMPLICATIONS FIGURE 11.4 PROOF. We may assume without loss of generality that D is bounded. Suppose nex ...

11.3. HOMOGENEITY AND QUASIDISKS I •mI J · FIGURE 11.5 D 155 Next let (3j and (3j be the components of 8D \ { Wj, wj} and choose ...

156 11. FOURTH SERIES OF IMPLICATIONS -2 - 2 let L j : R --+ R be a n affine mapping so that and let gj = Lj o fj for j = 1, 2,. ...

w" J 11.3. HOMOGENEITY AND QUASIDISKS / w'J. FIGURE 11. 6 157 then the points wj, w 1 , z 21 form triples which reduce the situa ...

158 11. FOURTH SERIES OF IMPLICATIONS with s > 0, so that z 1 , w, and w" are distinct points. We get a contradiction as befo ...

11.4. EXTREMAL DISTANCE DOMAINS PROOF. Choose z 1 E C1 and z2 E C2 so that lz1 - z2I = dist(C1, C2). Then by hypothesis there ex ...

160 11. FOURTH SERIES OF IMPLICATIONS Now let r 1 be the family of curves in r D which lie in B ( Z2, s) where (11.4.7) s = r (4 ...

11.4. EXTREMAL DISTANCE DOMAINS 161 such that i'1 joins 10 to a point w 2 E C with Jz2 - w2J ::; :2 lz1 - z2J. Continuing in thi ...

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164 BIBLIOGRAPHY [27] B. Brechner and T. Erkama, On topologically and quasiconformally homogeneous continua. Ann. Acad. Sci. Fen ...

BIBLIOGRAPHY 165 [59] F. W. Gehring and K. Hag, Reflections on reflections in quasidisks, Report. Univ. J yviiskyla 83 (2001) 81 ...

166 BIBLIOGRAPHY [89] D. S. Jerison and C. E. Kenig, Boundary behaviour of harmonic functions in non- tangentially accessible do ...

BIBLIOGRAPHY 167 [1 22 ] P. MacManus, R. Nakki, and B. Palka, Quasiconformally bi-homogeneous compacta in the complex plane. Pro ...

168 BIBLIOGRAPHY [156] D. Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. Anna ...