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in D. We have f+a on r and f+a“ on I
as not compromised too Notes to Chapter IX 339 grievously with esthetics. CarathCodory himself was critical of the cavalier proof which Koebe offered for his fundamental lemma 9.10 and gave (in [1913b]) another (KOEBE rejoins on pp. 212-213 of [1915]); that in the text is adapted from GOLUZIN [1969]. For another proof see MYLLER-LI~BBDEFF [1938], TSUJI [1959], p. 302 and for generalizations see 5 19 of MONTEL [1917],louis vuitton outlet store,FLEET[1954], Q 5 of BAGEMIHLand SEIDEL [1960], GAVRILOV [1960], [1961], BAGEMIHL [1965], CHOU [1967], RUNG [1968] and pp. 261-270 of POMMERENKE [1975]. A measure-theoretically more careful treatment of the main boundary argument in COURANT [1914] and FABER[I9221 may be found on pp. 308-311 of RUDIN [1974]. (See also NOVINGER [1975a].) There are several alternative proofs that g is univalent (lemma 9.13). See, e.g., chapter 6 of HEINS [1962], where the solubility of the Dirichlet problem is utilized. Monte1 and Lindelof use a version of 5.56 to this end: the half-disk in 5.56 is replaced with D = D(i, 1). However, in addition one needs to know that every point of aR is accessible from R (cf. 1,jordan 8 aqua.16). This is true, and trivial after the fact, i.e., after 9.14 is proved, but an ab initio proof is unpleasant,jordan shoes for sale. With this result in hand the proof of 9.13 is trivial: If a, a“ E iX2 and g(a) = w = g(cT), select arcs y, jj with terminal points a, 5,and lying, except for these points, in R,jordans outlet store. Then I’ = g 0 y, f = g 0 jj are arcs with common terminal point w which lie, except for this point, in D. We have f+a on r and f+a“ on I?, so by 5.56 d = a. The reader may note that the mapsf and g =f-l are treated by different tech- niques in the proof of the Osgood-Taylor-CarathCodory theorem,cheap jordan for sale. It should be clear, however, that if the arc lemma 9.10 is freed from the disk, then the technique of 9.11 could be used on g as well as f. This approach is offered in FLETT [1954]. It rests on an inequality of Carleman (see pp. 301-302 of TSUJI [1959]) and, even after the details are sup
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