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for sufficiently small , where c 4 and c 5 are positive absolute constants.Now for jxj 4 p 1 C 1=.4K/, we have ' 0 .x/ D 2.x x 0 / C O. 1 1=2 /, where x 0 WD 1=.2 p /;integrating this estimate, we obtain'.x/ D '.x 0 / .x x 0 / 2 C O. 1 1=2 / 3 4 C O. 1 1=2 /for jxfor jxx 0 j 1=2, and1 C O. 1 1=2 / j' 0 .x/j D O.1/; (2.44)x 0 j jxj 4 p 1 C 1=.4K/. Furthermore, it is easily checked thatr !' x 0 ˙ 2 1 C 1 1 C O. 1 1=2 /:4KFrom these estimates, we conclude that '.x/ has two zeroes x C D x C .; 2 ; 3 / and x D x .; 2 ; 3 /in the interval jxj 4 p 1 C 1=.4K/, such thatrx˙ ! ˙ WD x 0 ˙ 1 C 14 ; (2.45)uniformly for all K > 0 as ! 0.Since '.x C / D 0, we can integrate the estimate (2.44) and obtainZ xj'.x/j Dˇ ' 0 .t/ dtˇ jx C xj.1 C O. 1 1=2 //: (2.46)x CConsequently, there can be at most a finite number of m (indeed two if is sufficiently small) such thatj'.x/j 1 = and in that casejı n;m j˛˛ eˇˇˇe m '.x/=2m!ˇ1ˇ˛ D OOn the other hand, when j'.x/j 1 = and x satisfies (2.40), we havejı n;m j˛DD O˛ eˇˇˇe m '.x/=2m!˛ eˇˇˇe m '.x/=2m!˛ eˇˇˇe m '.x/=2Combining (2.47) and (2.48), we obtainXX ˛jı n;m j˛ De m '.x/=2ˇˇˇem!jxj.= 1 / 1=18 jxj.= 1 / 1=18m!C O 1D 2W˛./ C OXj'.x/j 1 = ˛˛1 3˛=2 : (2.47)ˇ1ˇ˛ ˇ1 .1 C x1ˇ˛10 ˛/1 C O1ˇ je '.x/=2 1j ˇˇ1ˇ˛ 1 1 .1 C x10 / : (2.48)ˇ1ˇ˛˛ eˇˇˇe m '.x/=2m! ˛1 .˛C1/=2 C ˛˛1 3˛=2 ;19! ˇ1ˇ˛ 1 C O ˛˛1 3˛=2