Content-Type: multipart/mixed; boundary="-------------0311071306432" This is a multi-part message in MIME format. ---------------0311071306432 Content-Type: text/plain; name="03-492.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-492.comments" E-mail adress Pierre.castillo@math.u-psud.fr AMS code 34A05, 35A 05 ---------------0311071306432 Content-Type: text/plain; name="03-492.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-492.keywords" Superconductivity, Ginzburg-Landau equations, critical field, Parr Formula ---------------0311071306432 Content-Type: application/postscript; name="parr3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="parr3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86f Copyright 2001 Radical Eye Software %%Title: parr3.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips parr3.dvi 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y(W)-8 b(e)29 b(consider)g(the)g(set)g Ff(H)842 5619 y Fe(1)944 5604 y Ff(\032)g Fj(I)-25 b(R)1143 5568 y Fg(+)1230 5604 y Fn(of)28 b(the)h Fj(h)p Fn('s)g(suc)m(h)i(that)d (there)h(exist)g(solutions)f(of)g(the)h(\()p Fj(G:L:)p Fn(\))g(sys-)-94 5725 y(tem)g(with)g Fj(f)38 b(>)28 b Fn(0)p Fj(:)i Fn(W)-8 b(e)30 b(kno)m(w)h(that)e Ff(H)1322 5740 y Fe(1)1427 5725 y Fn(is)g(a)g(b)s(ounded)i(in)m(terv)-5 b(al)28 b([0)p Fj(;)17 b(h)2521 5688 y Fg(+)2580 5725 y Fn(\))29 b(\(see)i([3],)f(Prop)s(osition)e(2.1\))-94 5845 y(and)k(w)m(e)i(then)f(in)m(tro)s(duce)g(the)g(follo)m(wing)c (de\014nition:)p eop %%Page: 2 2 2 1 bop 94 213 a Fn(2)94 511 y Fk(De\014nition)37 b(1.1)49 b Fc(The)42 b(sup)-5 b(erhe)g(ating)43 b(\014eld)f Fj(h)1858 475 y Fi(sh)1936 511 y Fn(\()p Fj(\024)p Fn(\))h Fc(is)g(de\014ne)-5 b(d)43 b(as)g(the)g(supr)-5 b(emum)43 b(of)g(the)g(inter-)94 632 y(val)35 b Ff(H)333 647 y Fe(1)408 632 y Fc(.)94 828 y Fn(In)e(the)g(w)m(eak-)p Fj(\024)h Fn(limit,)29 b(P)-8 b(.)33 b(G.)f(de)h(Gennes)h([16])e(has)h(prop)s(osed)g(the)g (form)m(ula)100 1054 y(lim)94 1111 y Fi(\024)p Fe(!)p Fg(0)258 1054 y Fj(\024)331 958 y Fh(\020)380 1054 y Fj(h)436 1013 y Fi(sh)514 1054 y Fn(\()p Fj(\024)p Fn(\))646 958 y Fh(\021)695 981 y Fg(2)762 1054 y Fn(=)28 b(2)915 1013 y Fe(\000)980 986 y Fd(3)p 979 998 31 4 v 979 1039 a(2)1024 1054 y Fj(:)2622 b Fn(\(1.4\))94 1278 y(In)33 b([3])g(and)g([5],)f(C.)h(Bolley)f(and)g(B.)h(Hel\013er)f(ha)m(v)m(e)i (pro)m(v)m(ed)g(rigorously)d(\(1.4\).)94 1399 y(H.)j(P)m(arr)f([19])g (has)h(prop)s(osed)f(on)g(the)h(basis)f(of)g(some)g(heuristic)f (computations,)h(the)g(more)g(general)94 1519 y(form)m(ula)94 1735 y Fj(\024)167 1639 y Fh(\020)217 1735 y Fj(h)273 1694 y Fi(sh)350 1735 y Fn(\()p Fj(\024)p Fn(\))482 1639 y Fh(\021)532 1662 y Fg(2)599 1735 y Fn(=)28 b(2)752 1694 y Fe(\000)817 1667 y Fd(3)p 816 1679 V 816 1720 a(2)883 1735 y Fn(+)991 1668 y(15)p 991 1712 98 4 v 991 1803 a(32)1099 1735 y Fj(\024)22 b Fn(+)g Fj(o)p Fn(\()p Fj(\024)p Fn(\))p Fj(:)2192 b Fn(\(1.5\))94 1963 y(Let)33 b(us)g(recall)f(that)g(w)m(e)i(ha)m(v)m(e)f(pro)m(v)m(ed)h(in)e([11])h (the)g(follo)m(wing)c(theorem:)94 2159 y Fk(Theorem)38 b(1.2)49 b Fc(Ther)-5 b(e)31 b(exist)g Fj(\024)1318 2174 y Fg(0)1386 2159 y Fj(>)c Fn(0)32 b Fc(and)f Fj(C)39 b Fc(such)32 b(that,)g(for)g(al)5 b(l)32 b Fj(\024)27 b Ff(\024)i Fj(\024)2845 2174 y Fg(0)2884 2159 y Fc(,)k(we)e(have)g (the)h(ine)-5 b(quality)94 2395 y Fj(\024)p Fn(\()p Fj(h)244 2354 y Fi(sh)322 2395 y Fn(\()p Fj(\024)p Fn(\)\))492 2354 y Fg(2)559 2395 y Ff(\025)28 b Fn(2)713 2354 y Fe(\000)778 2326 y Fd(3)p 778 2338 31 4 v 778 2380 a(2)845 2395 y Fn(+)953 2327 y(15)p 953 2371 98 4 v 953 2463 a(32)1060 2395 y Fj(\024)23 b Fn(+)f Fj(C)7 b(\024)1370 2354 y Fg(2)1409 2395 y Fj(:)2237 b Fn(\(1.6\))94 2617 y(In)29 b(this)g(pap)s(er,)g(to)g(complete)e(the)i(pro)s(of)f(of)g(the)h(P)m (arr)g(F)-8 b(orm)m(ula,)28 b(w)m(e)h(pro)m(v)m(e)h(the)f(follo)m(wing) d(theorem:)94 2813 y Fk(Theorem)38 b(1.3)49 b Fc(Ther)-5 b(e)27 b(exist)h Fj(\024)1311 2828 y Fg(0)1379 2813 y Fc(and)g Fj(\032)g(>)f Fn(0)p Fc(,)i(such)g(that,)g(for)f(al)5 b(l)28 b(p)-5 b(airs)28 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))28 b Fc(solution)g(of)g Fn(\()p Fj(GL)p Fn(\))3769 2828 y Fe(1)3844 2813 y Fc(,)94 2934 y(for)35 b(al)5 b(l)35 b Fj(\024)27 b Ff(\024)i Fj(\024)635 2949 y Fg(0)674 2934 y Fc(,)35 b(we)f(have)h(the)f(ine)-5 b(quality)94 3169 y Fj(\024A)223 3128 y Fe(0)247 3169 y Fn(\(0\))372 3128 y Fg(2)439 3169 y Fn(=)27 b Fj(\024h)654 3128 y Fg(2)722 3169 y Ff(\024)h Fn(2)876 3128 y Fe(\000)941 3101 y Fd(3)p 941 3113 31 4 v 941 3154 a(2)1007 3169 y Fn(+)1115 3102 y(15)p 1115 3146 98 4 v 1115 3237 a(32)1223 3169 y Fj(\024)22 b Fn(+)g Ff(O)s Fn(\()p Fj(\024)1575 3128 y Fg(1+)p Fi(\032)1706 3169 y Fn(\))p Fj(:)1902 b Fn(\(1.7\))94 3419 y Fk(Remark)37 b(1.4)49 b Fc(In)34 b([2],)h(C.)f(Bol)5 b(ley)34 b(and)g(the)h(author)g(have)f(pr)-5 b(ove)g(d)34 b(the)h(existenc)-5 b(e)33 b(and)h(the)h(unique-)94 3540 y(ness)46 b(of)g(a)g(solution)h(of)f Fn(\()p Fj(GL)p Fn(\))1268 3555 y Fe(1)1389 3540 y Fc(when)g(the)g(value)g(of)g Fj(f)11 b Fn(\(0\))48 b Ff(2)p Fn(]0)p Fj(;)17 b Fn(1])47 b Fc(is)f(given.)78 b(This)46 b(p)-5 b(ermits)46 b(to)94 3660 y(intr)-5 b(o)g(duc)g(e)47 b(the)g(map)f Fj(t)k Ff(7!)f Fj(\033)t Fn(\()p Fj(t)p Fn(\))e Fc(which)f(has)g(the)h(pr)-5 b(op)g(erty)47 b(that)g(if)g Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))47 b Fc(is)f(a)h(solution)f(of)h(the)94 3780 y(\(G.L.\))g(e)-5 b(quations)46 b(such)g(that)h Fj(f)11 b Fn(\(0\))48 b(=)h Fj(f)1706 3795 y Fg(0)1746 3780 y Fc(,)g(then)e Fj(A)2127 3744 y Fe(0)2150 3780 y Fn(\(0\))i(=)g Fj(\033)t Fn(\()p Fj(f)11 b Fn(\(0\)\))p Fj(:)46 b Fc(The)-5 b(or)g(em)45 b(1.3)h(p)-5 b(ermits)46 b(to)94 3901 y(obtain)35 b(an)f(upp)-5 b(er)35 b(b)-5 b(ound)35 b(for)f(the)h(maximum)f(of)g Fj(\033)39 b Fc(on)c Fn(]0)p Fj(;)17 b Fn(1])p Fc(.)94 4120 y Fn(In)41 b([5],)i(to)e(get)f(an)h(upp)s(er)g(b)s(ound)g(for)f Fj(A)1665 4084 y Fe(0)1688 4120 y Fn(\(0\))h(and)f(to)h(pro)m(v)m(e)h (the)f(de)g(Gennes)h(F)-8 b(orm)m(ula)38 b(\(1.4\),)k(C.)94 4240 y(Bolley)35 b(and)h(B.)g(Hel\013er)g(ha)m(v)m(e)h(pro)m(v)m(ed)g (the)f(follo)m(wing)e(estimates,)i(for)f(an)m(y)i(pairs)e(\()p Fj(f)5 b(;)17 b(A)p Fn(\))36 b(solutions)94 4361 y(of)d(\()p Fj(GL)p Fn(\))425 4376 y Fe(1)499 4361 y Fn(:)94 4557 y Fj(A)167 4516 y Fe(0)191 4557 y Fn(\(0\))316 4516 y Fg(2)383 4557 y Ff(\024)488 4470 y(p)p 571 4470 49 4 v 87 x Fn(2[\(1)22 b Ff(\000)g Fj(f)11 b Fn(\(0\))1039 4516 y Fg(2)1078 4557 y Fn(\))p Fj(f)g Fn(\(0\))1300 4516 y Fg(2)1339 4557 y Fj(\024)1395 4516 y Fe(\000)p Fg(1)1511 4557 y Fn(+)22 b(5)p Fj(A)1731 4516 y Fe(0)1755 4557 y Fn(\(0\))p Fj(f)11 b Fn(\(0\))2064 4516 y Fe(\000)p Fg(1)2157 4557 y Fn(])p Fj(;)1462 b Fn(\(1.8\))94 4777 y Fj(\024A)223 4735 y Fe(0)247 4777 y Fn(\(0\))372 4735 y Fg(2)439 4777 y Ff(\025)544 4690 y(p)p 627 4690 V 87 x Fn(2\(1)22 b Ff(\000)g Fj(f)11 b Fn(\(0\))1068 4735 y Fg(2)1107 4777 y Fn(\))p Fj(f)g Fn(\(0\))1329 4735 y Fg(2)1368 4777 y Fj(:)2278 b Fn(\(1.9\))94 5040 y(In)38 b([8])f(\(see)h(Prop)s(osition)d(3\),)i(in)g(order)g(to)f(pro)m(v)m(e)i (that)f(lim)2361 5056 y Fi(f)7 b Fg(\(0\))p Fe(!)p Fg(0)2619 5040 y Fj(\033)t Fn(\()p Fj(f)k Fn(\(0\)\))34 b(=)3134 4973 y(1)p 3093 5017 132 4 v 3093 5035 a Ff(p)p 3176 5035 49 4 v 82 x Fn(2)3235 5040 y(,)k(these)g(authors)94 5208 y(ha)m(v)m(e)c(pro)m(v)m(ed)g(the)f(follo)m(wing)d(estimate.)94 5405 y Fk(Prop)s(osition)36 b(1.5)49 b Fc(Ther)-5 b(e)33 b(exist)h Fj(\024)1453 5420 y Fg(0)1526 5405 y Fc(and)g Fj(C)41 b Fc(such)33 b(that,)i(for)e(al)5 b(l)34 b Fj(\024)28 b Ff(\024)g Fj(\024)2813 5420 y Fg(0)2886 5405 y Fc(and)34 b(any)f(solution)h Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))94 5525 y Fc(of)35 b Fn(\()p Fj(GL)p Fn(\))428 5540 y Fe(1)503 5525 y Fc(,)g(we)f(have)94 5777 y Fj(A)167 5735 y Fe(0)191 5777 y Fn(\(0\))316 5735 y Fg(2)383 5777 y Fn(=)27 b Fj(h)542 5735 y Fg(2)609 5777 y Ff(\024)725 5709 y Fn(1)p 725 5753 V 725 5845 a(2)805 5777 y(+)22 b Fj(C)990 5709 y(f)11 b Fn(\(0\))p 990 5753 184 4 v 1054 5845 a Fj(\024)1184 5777 y(:)2414 b Fn(\(1.10\))p eop %%Page: 3 3 3 2 bop 3636 213 a Fn(3)3 511 y(F)-8 b(rom)39 b(\(1.8\),)j(taking)d (the)i(maxim)m(um)d(of)i(\(1)27 b Ff(\000)h Fj(f)11 b Fn(\(0\))2006 475 y Fg(2)2045 511 y Fn(\))p Fj(f)g Fn(\(0\))2267 475 y Fg(2)2346 511 y Fn(on)40 b(the)h(in)m(terv)-5 b(al)39 b([0)p Fj(;)17 b Fn(1],)42 b(and)e(from)-94 632 y(\(1.10\),)31 b(they)j(ha)m(v)m(e)g(deduced)g(the)f(upp)s(erb)s(ound)-94 839 y Fj(\024h)18 798 y Fg(2)85 839 y Ff(\024)28 b Fn(2)239 798 y Fe(\000)304 771 y Fd(3)p 304 783 31 4 v 304 824 a(2)370 839 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)654 771 y Fd(1)p 655 783 V 655 824 a(2)699 839 y Fn(\))p Fj(;)2672 b Fn(\(1.11\))-94 1033 y(for)32 b(all)e Fj(h)e Ff(2)g(H)452 1048 y Fe(1)559 1033 y Fn(and)33 b(for)f Fj(\024)h Fn(small)d(enough.) -94 1227 y Fk(Remark)36 b(1.6)49 b Fc(F)-7 b(r)i(om)33 b(Pr)-5 b(op)g(osition)33 b(1.5,)h(we)f(de)-5 b(duc)g(e)33 b(that)i(Ine)-5 b(quality)33 b(\(1.7\))g(is)h(true)g(for)g(any)g(p)-5 b(airs)-94 1392 y Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))40 b Fc(solutions)g(of)h Fn(\()p Fj(GL)p Fn(\))946 1407 y Fe(1)1061 1392 y Fc(such)g(that)g Fj(f)11 b Fn(\(0\))37 b Ff(\024)1904 1325 y Fn(1)p 1841 1369 175 4 v 1841 1460 a(10)p Fj(C)2025 1392 y Fc(.)62 b(In)40 b(al)5 b(l)40 b(the)h(fol)5 b(lowing)39 b(se)-5 b(ctions,)41 b(we)g(wil)5 b(l)-94 1600 y(r)-5 b(estrict)35 b(us)g(to)g(the)g(p)-5 b(airs)34 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))35 b Fc(such)g(that)g Fj(f)11 b Fn(\(0\))27 b Ff(\025)1979 1533 y Fn(1)p 1916 1577 V 1916 1668 a(10)p Fj(C)2101 1600 y Fc(.)-94 1820 y Fn(The)i(plan)f(of)g(this)g(pap)s(er)h(is)f(the)h(follo)m(wing.)40 b(In)29 b(Section)f(2,)i(w)m(e)f(\014rst)g(recall)f(some)g(estimates)g (for)h(the)-94 1940 y(functions)36 b Fj(f)11 b Fn(,)37 b Fj(A)p Fn(,)g Fj(f)649 1904 y Fe(0)709 1940 y Fn(and)f Fj(A)975 1904 y Fe(0)1035 1940 y Fn(obtained)g(in)f([3])h(and)h([5].)55 b(Then,)38 b(w)m(e)g(impro)m(v)m(e)e(estimates)g(obtained)-94 2060 y(for)d Fj(A)g Fn(and)h Fj(A)426 2024 y Fe(0)449 2060 y Fn(.)47 b(In)33 b(Section)h(3,)g(w)m(e)g(analyze)g(the)g (estimate)e(for)h Fj(A)2376 2024 y Fe(0)2400 2060 y Fn(\(0\))c(=)g Fj(h)34 b Fn(obtained)f(b)m(y)h(C.)g(Bolley)-94 2181 y(and)j(B.)g(Hel\013er)f(in)h([5].)56 b(W)-8 b(e)37 b(get)g(a)g(b)s (etter)g(estimate)f(for)h Fj(f)2183 2145 y Fe(0)2243 2181 y Fn(in)f(the)h(in)m(terv)-5 b(al)36 b([0)p Fj(;)17 b(\024)3066 2145 y Fe(\000)p Fi(\032)3161 2181 y Fn(])p Fj(;)52 b(\032)35 b(>)g Fn(0.)57 b(In)-94 2301 y(Section)39 b(4,)j(using)e(a)g(principle)e(maxim)m(um,)i(w)m(e)h(impro)m(v)m(e)f (the)g(upp)s(er)h(b)s(ound)f(for)f Fj(A)3153 2265 y Fe(0)3177 2301 y Fn(\(0\))h(giv)m(en)g(in)-94 2422 y(\(1.8\))32 b(and)g(get)h(Theorem)g(1.3.)43 b(W)-8 b(e)33 b(deduce)h(the)f(P)m(arr) g(F)-8 b(orm)m(ula)30 b(\(1.5\).)-94 2658 y Fk(2.)56 b(Prop)s(erties)36 b(of)h(solutions)g(of)g Fn(\()p Fj(GL)p Fn(\))1539 2673 y Fe(1)-94 2856 y Fk(2.1.)56 b(General)37 b(prop)s(erties)g(of)h(the)f(Ginzburg-Landau)j(equations)3 2976 y Fn(Let)33 b(us)g(recall)e(some)i(prop)s(erties)f(on)h(the)g (functions)f Fj(f)11 b Fn(,)32 b Fj(A)p Fn(,)h Fj(f)2300 2940 y Fe(0)2356 2976 y Fn(and)f Fj(A)2618 2940 y Fe(0)2674 2976 y Fn(obtained)g(in)g([3])h(and)f([4].)-94 3192 y Fk(Prop)s(osition)j(2.1)49 b Fc(L)-5 b(et)35 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))35 b Fc(b)-5 b(e)35 b(a)f(solution)h(of)g Fn(\()p Fj(GL)p Fn(\))2059 3207 y Fe(1)2133 3192 y Fc(.)-94 3313 y Fn(1)p Fj(:)90 b(f)45 b Fc(is)35 b(incr)-5 b(e)g(asing)33 b(on)i Fj(I)-25 b(R)962 3276 y Fg(+)1055 3313 y Fc(and)35 b(we)f(have)-94 3524 y Fn(0)27 b Ff(\024)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\024)g Fn(tanh\()663 3457 y Fj(\024x)p 653 3501 132 4 v 653 3519 a Ff(p)p 736 3519 49 4 v 82 x Fn(2)817 3524 y(+)22 b Fj(x)970 3539 y Fg(0)1010 3524 y Fn(\))p Fj(;)99 b Fn(tanh)q(\()p Fj(x)1463 3539 y Fg(0)1502 3524 y Fn(\))28 b(=)f Fj(f)11 b Fn(\(0\))p Fj(:)1603 b Fn(\(2.1\))-94 3771 y(2)p Fj(:)90 b(A)35 b Fc(is)f(strictly)i(incr)-5 b(e)g(asing)33 b(on)i Fn([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)34 b Fc(and)g(we)h(have)-94 3965 y Fn(0)27 b Ff(\024)h Fj(A)160 3924 y Fe(0)184 3965 y Fn(\()p Fj(x)p Fn(\))g Ff(\024)g Fj(h;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)p Fj(:)2244 b Fn(\(2.2\))-94 4159 y(3)p Fj(:)90 b(f)131 4123 y Fe(0)188 4159 y Fc(satis\014es)35 b(the)g(ine)-5 b(qualities)-94 4371 y Fn(0)27 b Ff(\024)h Fj(f)146 4330 y Fe(0)169 4371 y Fn(\()p Fj(x)p Fn(\))g Ff(\024)481 4303 y Fj(\024)p 443 4347 132 4 v 443 4365 a Ff(p)p 526 4365 49 4 v 82 x Fn(2)585 4371 y Fj(;)127 b Ff(8)28 b Fj(x)h Ff(2)f Fn([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)p Fj(:)2135 b Fn(\(2.3\))-94 4618 y(4)p Fj(:)90 b Fc(The)34 b(p)-5 b(air)34 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))35 b Fc(satis\014es)f(the)h(fol)5 b(lowing)34 b(ener)-5 b(gy)34 b(c)-5 b(onservation)-94 4871 y Fj(\024)-38 4830 y Fe(\000)p Fg(2)56 4871 y Fj(f)115 4830 y Fe(0)138 4871 y Fn(\()p Fj(x)p Fn(\))269 4830 y Fg(2)331 4871 y Fn(+)22 b Fj(A)502 4830 y Fe(0)525 4871 y Fn(\()p Fj(x)p Fn(\))656 4830 y Fg(2)724 4871 y Fn(=)27 b Fj(A)p Fn(\()p Fj(x)p Fn(\))1031 4830 y Fg(2)1071 4871 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1261 4830 y Fg(2)1323 4871 y Fn(+)1431 4803 y(\(1)22 b Ff(\000)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1829 4767 y Fg(2)1869 4803 y Fn(\))1907 4767 y Fg(2)p 1431 4848 516 4 v 1664 4939 a Fn(2)1956 4871 y Fj(;)127 b Ff(8)28 b Fj(x)h Ff(2)f Fn([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)p Fj(:)764 b Fn(\(2.4\))-94 5096 y(5)p Fj(:)90 b Fc(The)34 b(p)-5 b(air)34 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))35 b Fc(satis\014es)f(the)h(ine)-5 b(qualities)-94 5290 y Fn(0)27 b Fj(<)h Ff(\000)p Fj(A)17 b(f)38 b Ff(\024)28 b Fj(A)517 5249 y Fe(0)541 5290 y Fj(;)127 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)p Fj(:)2180 b Fn(\(2.5\))-94 5506 y Fk(Remark)36 b(2.2)49 b Fc(F)-7 b(r)i(om)29 b(Pr)-5 b(op)g(osition)29 b(2.1)g(\(se)-5 b(e)28 b(\(2.1\)\),)i(we)f(de)-5 b(duc)g(e)29 b(that)h(in)f(a)h(r)-5 b(e)g(gion)28 b Fn([0)p Fj(;)17 b(\024)3242 5469 y Fe(\000)p Fi(\032)3337 5506 y Fn(])p Fc(,)31 b Fj(\032)d(>)f Fn(0)p Fc(,)-94 5626 y(we)34 b(have)-94 5820 y Fj(f)11 b Fn(\(0\))27 b Ff(\024)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))27 b Ff(\024)i Fj(f)11 b Fn(\(0\))21 b(+)h Ff(O)s Fn(\()p Fj(\024)1024 5779 y Fg(1)p Fe(\000)p Fi(\032)1155 5820 y Fn(\))p Fj(:)2265 b Fn(\(2.6\))p eop %%Page: 4 4 4 3 bop 94 213 a Fn(4)94 511 y Fk(2.2.)57 b(New)37 b(estimates)f(for)i Fj(f)1309 475 y Fe(0)1332 511 y Fk(,)f Fj(A)h Fk(and)g Fj(A)1800 475 y Fe(0)192 632 y Fn(In)c(the)h(next)g(sections,)g(w)m(e)g (use)g(t)m(w)m(o)g(useful)f(v)m(ersions)h(of)f(the)g(maxim)m(um)e (principle)h(\(cf.)48 b([5])34 b(and)94 752 y([7]\).)94 949 y Fk(Lemma)k(2.3)49 b Fc(L)-5 b(et)35 b Fj(d)27 b Ff(2)h Fj(I)-25 b(R)1117 913 y Fg(+)1198 949 y Ff([)23 b(f)p Fn(+)p Ff(1g)p Fc(.)44 b(L)-5 b(et)35 b Fj(C)42 b Fc(b)-5 b(e)34 b(a)h(b)-5 b(ounde)g(d)34 b(function)h(on)f Fn(]0)p Fj(;)17 b(d)p Fn([)34 b Fc(such)h(that)1511 1161 y Fj(C)7 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\025)g Fn(0)p Fj(;)72 b Ff(8)28 b Fj(x)g Ff(2)p Fn(]0)p Fj(;)17 b(d)p Fn([)p Fj(;)94 1373 y Fc(and)35 b(let)g Fj(u)27 b Ff(2)h Fj(C)675 1337 y Fg(2)714 1373 y Fn(\([0)p Fj(;)17 b(d)p Fn(]\))35 b Fc(b)-5 b(e)34 b(a)h(function)f(such)h(that)94 1486 y Fh(\()203 1571 y Ff(\000)p Fj(u)336 1535 y Fe(00)400 1571 y Fn(+)22 b Fj(C)7 b(u)28 b Ff(\024)g Fn(0)55 b Fc(on)g Fn(]0)p Fj(;)17 b(d)p Fn([)p Fj(;)203 1691 y(u)259 1655 y Fe(0)282 1691 y Fn(\(0\))27 b Ff(\025)h Fn(0)p Fj(:)3673 1632 y Fn(\(2.7\))94 1893 y Fc(If)35 b Fj(d)27 b Ff(2)h Fj(I)-25 b(R)462 1856 y Fg(+)521 1893 y Fc(,)35 b(we)f(assume)g(that)i Fj(u)p Fn(\()p Fj(d)p Fn(\))26 b Ff(\024)j Fn(0)p Fc(,)34 b(and)g(if)h Fj(d)27 b Fn(=)h(+)p Ff(1)p Fc(,)34 b Fj(u)2461 1856 y Fe(0)2512 1893 y Ff(!)27 b Fn(0)35 b Fc(when)f Fj(x)28 b Ff(!)f Fn(+)p Ff(1)p Fc(.)94 2013 y(Then,)34 b(in)h(these)g(two)f(c)-5 b(ases,)1646 2133 y Fj(u)27 b Ff(\024)h Fn(0)55 b Fc(on)g Fn(]0)p Fj(;)17 b(d)p Fn([)p Fj(:)94 2330 y Fn(Let)33 b(us)g(recall)f(that,)g (from)f(Remark)i(2.2,)f(w)m(e)h(assume)g(that)94 2566 y Fj(f)11 b Fn(\(0\))28 b Ff(\025)484 2498 y Fn(1)p 421 2542 175 4 v 421 2634 a(10)p Fj(C)605 2566 y(:)3041 b Fn(\(2.8\))94 2792 y(Some)35 b(inequalities)e(on)i(the)h(functions)f Fj(f)1663 2755 y Fe(0)1686 2792 y Fn(,)h Fj(A)f Fn(and)g Fj(A)2122 2755 y Fe(0)2181 2792 y Fn(are)g(not)g(optimal.)48 b(In)35 b(this)g(section,)h(w)m(e)g(get)94 2912 y(a)42 b(b)s(etter)g(con)m(trol)e(on)i Fj(A)f Fn(and)h Fj(A)1351 2876 y Fe(0)1374 2912 y Fn(.)70 b(Let)42 b(us)g(recall)e(that)i(in)e ([3],)k(C.)e(Bolley)e(and)i(B.)f(Hel\013er)h(ha)m(v)m(e)94 3032 y(obtained)33 b(the)g(follo)m(wing)c(con)m(trol)j(for)g Fj(A)94 3273 y(h)17 b Fn(exp)q(\()p Ff(\000)p Fj(x)p Fn(\))29 b Ff(\024)f(\000)p Fj(A)g Ff(\024)1015 3205 y Fj(h)p 951 3250 184 4 v 951 3341 a(f)11 b Fn(\(0\))1161 3273 y(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fj(I)-25 b(R)2182 3232 y Fg(+)2241 3273 y Fj(:)1405 b Fn(\(2.9\))94 3525 y(The)37 b(comparison)d(with)h(the)h(b)s(eha)m(vior)f(of)g(the)h (formal)d(solution)h(giv)m(en)i(in)e([12])h(suggests)i(that)f(the)94 3646 y(lo)m(w)m(er)c(b)s(ound)f(for)f Ff(\000)p Fj(A)i Fn(is)e(not)h(optimal.)40 b(W)-8 b(e)32 b(get)f(a)f(b)s(etter)i(lo)m(w) m(er)f(b)s(ound)g(for)f Ff(\000)p Fj(A)i Fn(in)e(the)i(follo)m(wing)94 3766 y(prop)s(osition:)94 3963 y Fk(Prop)s(osition)k(2.4)49 b Fc(L)-5 b(et)38 b Fj(\032)33 b Ff(2)p Fn(]0)p Fj(;)1342 3924 y Fg(1)p 1342 3940 36 4 v 1342 3997 a(2)1387 3963 y Fn([)p Fc(.)52 b(Ther)-5 b(e)37 b(exist)g Fj(\024)2064 3978 y Fg(0)2142 3963 y Fc(and)f Fj(C)k(>)32 b Fn(0)37 b Fc(such)h(that,)g(for)f(al)5 b(l)37 b Fj(\024)c Ff(\024)g Fj(\024)3649 3978 y Fg(0)3689 3963 y Fj(;)k Fc(for)94 4083 y(al)5 b(l)33 b(p)-5 b(airs)32 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))33 b Fc(solutions)f(of)g Fn(\()p Fj(GL)p Fn(\))1486 4098 y Fe(1)1594 4083 y Fc(satisfying)g(\(2.8\),)g(the)h (function)g Fj(A)f Fc(satis\014es)g(the)h(ine)-5 b(quality)94 4329 y Ff(8)29 b Fj(x)55 b Ff(2)h Fn([0)p Fj(;)17 b(\024)586 4288 y Fe(\000)p Fi(\032)681 4329 y Fn(])p Fj(;)72 b Ff(\000)p Fj(A)28 b Ff(\025)1164 4262 y Fj(h)p 1101 4306 184 4 v 1101 4398 a(f)11 b Fn(\(0\))1294 4329 y(\(1)22 b Ff(\000)g Fj(C)7 b(\024)1635 4288 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)1801 4329 y Fn(\))17 b(exp)g Ff(\000)p Fn(\()p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)1185 b Fn(\(2.10\))94 4576 y Fk(Pro)s(of)94 4697 y Fn(W)-8 b(e)33 b(set:)94 4893 y Fj(v)141 4908 y Fg(2)181 4893 y Fn(\()p Fj(x)p Fn(\))28 b(:=)g(tanh\()724 4826 y Fj(\024x)p 714 4870 132 4 v 714 4888 a Ff(p)p 797 4888 49 4 v 82 x Fn(2)878 4893 y(+)22 b Fj(x)1031 4908 y Fg(0)1070 4893 y Fn(\))p Fj(;)100 b Fn(tanh\()p Fj(x)1523 4908 y Fg(0)1563 4893 y Fn(\))28 b(=)f Fj(f)11 b Fn(\(0\))p Fj(:)1682 b Fn(\(2.11\))94 5143 y(F)-8 b(rom)32 b(\(2.1\),)g(w)m(e)h(get)94 5340 y Fj(f)39 b Ff(\024)28 b Fj(v)333 5355 y Fg(2)373 5340 y Fj(:)3225 b Fn(\(2.12\))94 5536 y(Let)33 b(us)g(consider)g(the)g (unique)g(solution)e(in)h Fj(H)1831 5500 y Fg(2)1870 5536 y Fn(\(]0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([\))32 b(of)g(the)h (problem)94 5649 y Fh(\()203 5734 y Ff(\000)p Fj(W)386 5698 y Fe(00)451 5734 y Fn(+)22 b Fj(v)600 5698 y Fg(2)596 5759 y(2)639 5734 y Fj(W)41 b Fn(=)28 b(0)p Fj(;)203 5854 y(W)309 5818 y Fe(0)332 5854 y Fn(\(0\))f(=)h Fj(h:)3625 5795 y Fn(\(2.13\))p eop %%Page: 5 5 5 4 bop 3636 213 a Fn(5)-94 511 y(F)-8 b(rom)31 b(\(2.13\),)i(and)g (applying)f(Lemma)g(2.3)g(with)h Fj(d)c Fn(=)f(+)p Ff(1)p Fn(,)33 b Fj(C)j Fn(=)28 b Fj(v)2471 475 y Fg(2)2467 536 y(2)2543 511 y Fn(and)34 b Fj(u)28 b Fn(=)g Fj(W)14 b Fn(,)33 b(w)m(e)i(get)e Fj(W)42 b Ff(\024)29 b Fn(0)-94 632 y(on)34 b Fj(I)-25 b(R)136 596 y Fg(+)195 632 y Fn(.)51 b(F)-8 b(rom)33 b(\(1)p Fj(:)p Fn(2\))732 647 y Fi(b)766 632 y Fn(,)i(\(2.12\))g(and)g(\(2.13\),)f(w)m(e)i(get)f Ff(\000)p Fn(\()p Fj(A)25 b Ff(\000)f Fj(W)14 b Fn(\))2385 596 y Fe(00)2450 632 y Fn(+)24 b Fj(f)2609 596 y Fg(2)2648 632 y Fn(\()p Fj(A)g Ff(\000)g Fj(W)14 b Fn(\))31 b Ff(\024)i Fn(0.)50 b(Applying)-94 752 y(again)31 b(Lemma)g(2.3)h(with)g Fj(d)c Fn(=)f(+)p Ff(1)p Fn(,)32 b Fj(C)j Fn(=)27 b Fj(f)1577 716 y Fg(2)1649 752 y Fn(and)33 b Fj(u)27 b Fn(=)g Fj(A)c Ff(\000)f Fj(W)14 b Fn(,)33 b(w)m(e)g(get)-94 954 y Fj(A)27 b Ff(\024)h Fj(W)m(:)3209 b Fn(\(2.14\))-94 1157 y(W)-8 b(e)32 b(set)-94 1359 y Fj(\036)p Fn(\()p Fj(x)p Fn(\))27 b(:=)253 1242 y Fh(Z)336 1268 y Fi(x)299 1430 y Fg(0)380 1359 y Fn(\()p Fj(v)465 1374 y Fg(2)526 1359 y Fn(+)22 b Fj(c)666 1374 y Fg(1)706 1359 y Fj(\024)p Fn(\))p Fj(dt;)2523 b Fn(\(2.15\))-94 1590 y(where)33 b Fj(c)229 1605 y Fg(1)301 1590 y Fn(is)f(a)h(strictly)e(p)s(ositiv)m(e)h(parameter,)h(whic)m(h)g (will)d(b)s(e)j(determined)f(later.)43 b(W)-8 b(e)33 b(consider)-94 1792 y Fj(z)t Fn(\()p Fj(x)p Fn(\))28 b(=)g Fj(z)t Fn(\(0\))17 b(exp)g(\()p Ff(\000)p Fj(\036)p Fn(\()p Fj(x)p Fn(\)\))g Fj(;)2476 b Fn(\(2.16\))-94 1994 y(where)29 b Fj(z)t Fn(\(0\))f(is)f(determined)h(b)m(y)h(the)f (condition)e Fj(z)1749 1958 y Fe(0)1773 1994 y Fn(\(0\))i(=)f Fj(h)p Fn(.)42 b(W)-8 b(e)28 b(get)g Fj(z)2524 1958 y Fe(0)2548 1994 y Fn(\(0\))g(=)f Ff(\000)p Fn(\()p Fj(v)2966 2009 y Fg(2)3006 1994 y Fn(\(0\))12 b(+)g Fj(c)3273 2009 y Fg(1)3313 1994 y Fj(\024)p Fn(\))p Fj(z)t Fn(\(0\))28 b(=)-94 2114 y Fj(h)p Fn(,)k(hence,)i(from)e(\(2.11\))-94 2366 y Fj(z)t Fn(\(0\))c(=)f Ff(\000)491 2298 y Fj(h)p 298 2343 442 4 v 298 2434 a(f)11 b Fn(\(0\))22 b(+)g Fj(c)644 2449 y Fg(1)684 2434 y Fj(\024)750 2366 y(:)2659 b Fn(\(2.17\))-94 2618 y(W)-8 b(e)32 b(ha)m(v)m(e)i(the)f(equalit)m(y) -53 2812 y Ff(\000)p Fn(\()p Fj(W)j Ff(\000)23 b Fj(z)t Fn(\))377 2776 y Fe(00)442 2812 y Fn(+)f Fj(v)591 2776 y Fg(2)587 2837 y(2)630 2812 y Fn(\()p Fj(W)36 b Ff(\000)23 b Fj(z)t Fn(\))28 b(=)g Fj(z)t Fn(\(0\))17 b(exp)q(\()p Ff(\000)p Fj(\036)p Fn(\()p Fj(x)p Fn(\)\)\()p Ff(\000)1949 2773 y Fi(\024)p 1922 2789 95 4 v 1922 2798 a Fe(p)p 1981 2798 36 3 v 55 x Fg(2)2245 2773 y(1)p 2036 2789 453 4 v 2036 2850 a(cosh)2170 2824 y Fd(2)2205 2850 y Fg(\()2247 2823 y Fb(\024x)p 2242 2835 81 4 v 2242 2843 a Fa(p)p 2292 2843 31 3 v 42 x Fd(2)2332 2850 y Fg(+)p Fi(x)2427 2859 y Fd(0)2462 2850 y Fg(\))2521 2812 y Fn(+)22 b(2)p Fj(v)2715 2827 y Fg(2)2754 2812 y Fj(c)2796 2827 y Fg(1)2836 2812 y Fj(\024)g Fn(+)g Fj(c)3054 2776 y Fg(2)3054 2837 y(1)3094 2812 y Fj(\024)3150 2776 y Fg(2)3189 2812 y Fn(\))p Fj(:)3436 2834 y Fn(\(2.18\))-94 3050 y(F)-8 b(rom)31 b(the)i(de\014nition)e(of)h(the)h(function)f Fj(v)1471 3065 y Fg(2)1511 3050 y Fn(,)h(w)m(e)g(ha)m(v)m(e)h(the)f (inequalit)m(y)-53 3245 y Ff(\000)64 3206 y Fg(1)p 34 3222 95 4 v 34 3231 a Fe(p)p 93 3231 36 3 v 56 x Fg(2)155 3245 y Fn(cosh)340 3202 y Fe(\000)p Fg(2)434 3245 y Fn(\()489 3206 y Fi(\024x)p 482 3222 95 4 v 482 3231 a Fe(p)p 541 3231 36 3 v 56 x Fg(2)608 3245 y Fn(+)22 b Fj(x)761 3260 y Fg(0)801 3245 y Fn(\))g(+)g(2)p Fj(v)1055 3260 y Fg(2)1095 3245 y Fj(c)1137 3260 y Fg(1)1198 3245 y Fn(+)g Fj(c)1338 3209 y Fg(2)1338 3270 y(1)1378 3245 y Fj(\024)28 b Ff(\025)g Fn(2)p Fj(c)1658 3260 y Fg(1)1713 3245 y Fn(tanh)q(\()p Fj(x)2002 3260 y Fg(0)2041 3245 y Fn(\))22 b Ff(\000)2240 3206 y Fg(1)p 2211 3222 95 4 v 2211 3231 a Fe(p)p 2270 3231 36 3 v 56 x Fg(2)2332 3245 y Fn(cosh)2516 3202 y Fe(\000)p Fg(2)2611 3245 y Fn(\()p Fj(x)2704 3260 y Fg(0)2744 3245 y Fn(\))p Fj(:)3436 3252 y Fn(\(2.19\))-94 3454 y(W)-8 b(e)32 b(c)m(ho)s(ose)i Fj(c)425 3469 y Fg(1)497 3454 y Fn(suc)m(h)g(that)-94 3691 y Fj(c)-52 3706 y Fg(1)15 3691 y Fj(>)194 3624 y Fn(1)p 128 3668 181 4 v 128 3768 a(2)177 3686 y Ff(p)p 260 3686 49 4 v 82 x Fn(2)335 3691 y(\(sinh\()p Fj(x)640 3706 y Fg(0)680 3691 y Fn(\))17 b(cosh\()p Fj(x)1012 3706 y Fg(0)1052 3691 y Fn(\)\))1128 3643 y Fe(\000)p Fg(1)1239 3691 y Fj(:)2170 b Fn(\(2.20\))-94 3946 y(F)-8 b(rom)29 b(\(2.17\),)h(w)m(e)i(ha)m(v)m(e)g Fj(z)t Fn(\(0\))c Fj(<)f Fn(0)k(for)f Fj(\024)g Fn(small)f(enough.)43 b(F)-8 b(rom)29 b(\(2.18\),)h(\(2.19\))g(and)h(\(2.20\),)f(w)m(e)i(get) 1174 4165 y Ff(\000)p Fn(\()p Fj(W)k Ff(\000)22 b Fj(z)t Fn(\))1603 4124 y Fe(00)1669 4165 y Fn(+)g Fj(v)1818 4124 y Fg(2)1814 4189 y(2)1857 4165 y Fn(\()p Fj(W)36 b Ff(\000)22 b Fj(z)t Fn(\))29 b Fj(<)e Fn(0)p Fj(:)-94 4383 y Fn(Applying)g(Lemma)h(2.3)g(on)h([0)p Fj(;)17 b Fn(+)p Ff(1)p Fn([)28 b(with)g Fj(C)34 b Fn(=)28 b Fj(v)1782 4347 y Fg(2)1778 4408 y(2)1850 4383 y Fn(and)h Fj(u)e Fn(=)g Fj(W)h Ff(\000)14 b Fj(z)t Fn(,)31 b(w)m(e)f(deduce)h (that)d Fj(W)g Ff(\000)14 b Fj(u)28 b Ff(\024)g Fn(0.)-94 4504 y(F)-8 b(rom)31 b(\(2.16\))g(and)i(\(2.17\),)f(it)g(results)h (that)-94 4755 y Ff(\000)p Fj(W)41 b Ff(\025)28 b(\000)p Fj(z)33 b Fn(=)682 4688 y Fj(h)p 490 4732 442 4 v 490 4824 a(f)11 b Fn(\(0\))21 b(+)h Fj(c)835 4839 y Fg(1)875 4824 y Fj(\024)958 4755 y Fn(exp)q(\()p Ff(\000)p Fj(\036)p Fn(\()p Fj(x)p Fn(\)\))p Fj(:)1960 b Fn(\(2.21\))-94 5013 y(F)-8 b(rom)31 b(\(2.15\),)g(there)j(exists)f Fj(d)1040 5028 y Fi(x)1111 5013 y Ff(2)p Fn(]0)p Fj(;)17 b Fn(1[)33 b(suc)m(h)h(that)e(w)m(e)i(ha)m(v)m(e)g(the)f(equalit)m(y)1211 5287 y Fj(\036)p Fn(\()p Fj(x)p Fn(\))27 b(=)h Fj(\036)1589 5246 y Fe(0)1612 5287 y Fn(\(0\))p Fj(x)22 b Fn(+)1922 5219 y Fj(\036)1980 5183 y Fe(00)2022 5219 y Fn(\()p Fj(d)2111 5234 y Fi(x)2155 5219 y Fj(x)p Fn(\))p 1922 5264 327 4 v 2061 5355 a(2)2258 5287 y Fj(x)2313 5246 y Fg(2)2353 5287 y Fj(:)-94 5536 y Fn(According)32 b(to)g(\(2.11\))g (and)h(\(2.15\),)e(w)m(e)j(deduce)g(that)718 5804 y Fj(\036)p Fn(\()p Fj(x)p Fn(\))27 b(=)h(\()p Fj(f)11 b Fn(\(0\))21 b(+)h Fj(c)1421 5819 y Fg(1)1461 5804 y Fj(\024)p Fn(\))p Fj(x)g Fn(+)1782 5737 y(1)p 1740 5781 132 4 v 1740 5799 a Ff(p)p 1823 5799 49 4 v 82 x Fn(2)1899 5804 y(cosh)2084 5762 y Fe(\000)p Fg(2)2178 5804 y Fn(\()2226 5737 y Fj(d)2277 5752 y Fi(x)2320 5737 y Fj(\024x)p 2226 5781 207 4 v 2263 5799 a Ff(p)p 2346 5799 49 4 v 82 x Fn(2)2464 5804 y(+)g Fj(x)2617 5819 y Fg(0)2657 5804 y Fn(\))p Fj(\024x)2806 5763 y Fg(2)2846 5804 y Fj(:)p eop %%Page: 6 6 6 5 bop 94 213 a Fn(6)94 511 y(It)33 b(results)g(that)f(there)i(exists) f Fj(\032)28 b(>)g Fn(0)k(suc)m(h)i(that)94 764 y Ff(8)29 b Fj(x)f Ff(2)55 b Fn([0)p Fj(;)17 b(\024)558 723 y Fe(\000)p Fi(\032)653 764 y Fn(])p Fj(;)72 b(\036)p Fn(\()p Fj(x)p Fn(\))28 b Ff(\024)g Fn(\()p Fj(f)11 b Fn(\(0\))22 b(+)g Fj(c)1485 779 y Fg(1)1524 764 y Fj(\024)p Fn(\))p Fj(x)h Fn(+)1804 696 y Fj(\024)1860 660 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)p 1804 741 222 4 v 1849 758 a Ff(p)p 1932 758 49 4 v 82 x Fn(2)2036 764 y Fj(:)1562 b Fn(\(2.22\))94 1015 y(Finally)-8 b(,)30 b(from)i(\(2.14\),)g(\(2.21\))f(and)i (\(2.22\),)f(w)m(e)i(get)e(the)h(inequalit)m(y)94 1276 y Ff(\000)p Fj(A)c Ff(\025)f(\000)p Fj(W)42 b Ff(\025)896 1209 y Fj(h)p 704 1253 442 4 v 704 1345 a(f)11 b Fn(\(0\))21 b(+)h Fj(c)1049 1360 y Fg(1)1089 1345 y Fj(\024)1172 1276 y Fn(exp)1337 1130 y Fh( )1403 1276 y Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)22 b Ff(\000)h Fj(c)1883 1291 y Fg(1)1922 1276 y Fj(\024x)g Ff(\000)2166 1209 y Fj(\024)2222 1173 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)p 2166 1253 222 4 v 2211 1271 a Ff(p)p 2294 1271 49 4 v 82 x Fn(2)2398 1130 y Fh(!)2480 1276 y Fj(:)1118 b Fn(\(2.23\))94 1557 y(W)-8 b(e)33 b(deduce)i(the)e(existence)h(of)e(some)1548 1532 y(^)1526 1557 y Fj(C)j(>)27 b Fn(0,)33 b(suc)m(h)h(that,)e(for)g Fj(\024)h Fn(small)d(enough)94 1799 y Ff(\000)p Fj(A)f Ff(\025)452 1732 y Fj(h)p 388 1776 184 4 v 388 1868 a(f)11 b Fn(\(0\))581 1799 y(\(1)22 b Ff(\000)h Fj(c)832 1814 y Fg(1)945 1732 y Fj(\024)p 881 1776 V 881 1868 a(f)11 b Fn(\(0\))1074 1799 y(\))17 b(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))17 b(exp)q(\()p Ff(\000)1973 1774 y Fn(^)1951 1799 y Fj(C)7 b(\024)2084 1758 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)2250 1799 y Fn(\))p Fj(:)1310 b Fn(\(2.24\))94 2054 y(F)-8 b(rom)47 b(\(2.8\))g(and)g(\(2.24\),)k(w)m (e)d(deduce)i(that)d(there)h(exist)g Fj(C)60 b(>)53 b Fn(0)47 b(and)h Fj(\024)2997 2069 y Fg(0)3037 2054 y Fn(,)j(suc)m(h)e(that,)i(for)c(all)94 2174 y Fj(\024)28 b Ff(2)p Fn(]0)p Fj(;)17 b(\024)420 2189 y Fg(0)460 2174 y Fn(],)33 b(for)f(all)e Fj(x)e Ff(2)g Fn([0)p Fj(;)17 b(\024)1184 2138 y Fe(\000)p Fi(\032)1279 2174 y Fn(],)33 b(w)m(e)h(ha)m(v)m(e)f(the)g(inequalit)m(y)94 2422 y Ff(\000)p Fj(A)c Ff(\025)452 2355 y Fj(h)p 388 2399 V 388 2491 a(f)11 b Fn(\(0\))581 2422 y(\(1)22 b Ff(\000)h Fj(C)7 b(\024)923 2381 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)1089 2422 y Fn(\))17 b(exp)g(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))16 b Fj(:)1881 b Fn(\(2.25\))94 2671 y(The)34 b(pro)s(of)e(of)g(Prop)s(osition)f(2.4)h(follo)m(ws.)94 2912 y(W)-8 b(e)42 b(ha)m(v)m(e)h(also)e(to)g(get)g(a)h(lo)m(w)m(er)f (b)s(ound)h(and)f(an)h(upp)s(er)g(b)s(ound)g(for)e Fj(A)2844 2876 y Fe(0)2868 2912 y Fn(.)70 b(In)42 b([3],)i(C.)e(Bolley)e(and)94 3032 y(B.)33 b(Hel\013er)f(get)h(in)f(Prop)s(osition)f(2.4)h(the)h (follo)m(wing)d(estimate)i(for)g Fj(A)2689 2996 y Fe(0)2712 3032 y Fn(:)94 3275 y Fj(h)17 b Fn(exp)q(\()p Ff(\000)p Fj(x)p Fn(\))29 b Ff(\024)f Fj(A)731 3234 y Fe(0)754 3275 y Fn(\()p Fj(x)p Fn(\))g Ff(\024)1092 3208 y Fj(h)p 1028 3252 V 1028 3343 a(f)11 b Fn(\(0\))1238 3275 y(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)h Fj(I)-25 b(R)2260 3234 y Fg(+)2318 3275 y Fj(:)1280 b Fn(\(2.26\))94 3530 y(The)49 b(construction)f(of)f (a)h(formal)d(solution)h(in)h([12])g(suggests)i(that)f(these)h (inequalities)c(are)j(not)94 3650 y(optimal.)41 b(W)-8 b(e)33 b(get)g(a)f(b)s(etter)h(con)m(trol)f(for)g Fj(A)1742 3614 y Fe(0)1798 3650 y Fn(in)g(the)h(follo)m(wing)c(prop)s(osition:)94 3849 y Fk(Prop)s(osition)36 b(2.5)49 b Fc(L)-5 b(et)43 b Fj(\032)g Ff(2)g Fn([0)p Fj(;)1399 3809 y Fg(1)p 1399 3825 36 4 v 1399 3883 a(4)1445 3849 y Fn(])p Fc(.)68 b(Ther)-5 b(e)43 b(exists)f Fj(\024)2189 3864 y Fg(0)2272 3849 y Fc(such)g(that,)j(for)e(al)5 b(l)42 b Fj(\024)h Ff(\024)g Fj(\024)3326 3864 y Fg(0)3408 3849 y Fc(and)g(for)f(al)5 b(l)94 3969 y Fn(\()p Fj(f)g(;)17 b(A)p Fn(\))35 b Fc(solutions)g(of)f Fn(\()p Fj(GL)p Fn(\))1117 3984 y Fe(1)1227 3969 y Fc(satisfying)h (\(2.8\),)f(we)g(have)g(the)h(fol)5 b(lowing)34 b(estimate)g(for)h Fj(A)3496 3933 y Fe(0)3519 3969 y Fc(:)94 4181 y Fj(A)167 4140 y Fe(0)219 4181 y Fn(=)322 4084 y Fh(\020)372 4181 y Fj(h)22 b Fn(+)g Ff(O)s Fn(\()p Fj(\024)734 4113 y Fd(1)p 734 4125 31 4 v 734 4166 a(2)775 4140 y Fe(\000)p Fg(2)p Fi(\032)905 4181 y Fn(\))943 4084 y Fh(\021)1009 4181 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)17 b Fj(x)28 b Ff(2)g Fn([0)p Fj(;)17 b(\024)2102 4140 y Fe(\000)p Fi(\032)2197 4181 y Fn(])p Fj(:)1374 b Fn(\(2.27\))94 4379 y Fk(Pro)s(of)94 4500 y Fn(F)-8 b(rom)32 b(\(1)p Fj(:)p Fn(2\))552 4515 y Fi(b)585 4500 y Fn(,)h(the)g(function)f Fj(A)1268 4464 y Fe(0)1324 4500 y Fn(satis\014es)94 4698 y Ff(\000)p Fn(\()p Fj(A)282 4657 y Fe(0)306 4698 y Fn(\))344 4657 y Fe(00)409 4698 y Fn(+)22 b Fj(f)566 4657 y Fg(2)605 4698 y Fj(A)678 4657 y Fe(0)729 4698 y Fn(=)28 b Ff(\000)p Fn(2)p Fj(f)1018 4657 y Fe(0)1041 4698 y Fj(f)11 b(A;)127 b(A)1400 4657 y Fe(00)1443 4698 y Fn(\(0\))27 b(=)h Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))2081 4657 y Fg(2)2119 4698 y Fj(:)1479 b Fn(\(2.28\))94 4897 y(F)-8 b(rom)32 b(\(2.1\),)g(\(2.3\)) g(and)g(\(2.9\),)h(w)m(e)g(get)g(the)g(inequalit)m(y)94 5145 y Ff(\000)p Fn(2)p Fj(f)279 5104 y Fe(0)303 5145 y Fj(f)11 b(A)27 b Ff(\024)654 5078 y Fn(2)p Fj(\024h)p 577 5122 316 4 v 577 5140 a Ff(p)p 660 5140 49 4 v 82 x Fn(2)p Fj(f)11 b Fn(\(0\))919 5145 y(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fj(I)-25 b(R)1940 5104 y Fg(+)1999 5145 y Fj(:)1599 b Fn(\(2.29\))94 5412 y(W)-8 b(e)33 b(can)g(compare)f Fj(A)904 5376 y Fe(0)960 5412 y Fn(with)h(the)g(unique)g(solution)e(in) h Fj(H)2240 5376 y Fg(2)2278 5412 y Fn(\()p Fj(I)-25 b(R)2409 5376 y Fg(+)2468 5412 y Fn(\))33 b(of)94 5535 y Fh(8)94 5610 y(>)94 5635 y(<)94 5784 y(>)94 5809 y(:)210 5681 y Ff(\000)p Fj(y)339 5645 y Fe(00)403 5681 y Fn(+)22 b Fj(f)11 b Fn(\(0\))685 5645 y Fg(2)724 5681 y Fj(y)31 b Fn(=)916 5532 y Ff(p)p 999 5532 V 82 x Fn(2)p Fj(\024h)p 916 5658 245 4 v 946 5749 a(f)11 b Fn(\(0\))1187 5681 y(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)210 5859 y(y)262 5822 y Fe(0)284 5859 y Fn(\(0\))28 b(=)f Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))922 5822 y Fg(2)961 5859 y Fj(;)3625 5734 y Fn(\(2.30\))p eop %%Page: 7 7 7 6 bop 3636 213 a Fn(7)-94 511 y(giv)m(en)32 b(b)m(y)-94 734 y Fj(y)t Fn(\()p Fj(x)p Fn(\))27 b(=)219 588 y Fh( )285 734 y Ff(\000)p Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))22 b(+)874 584 y Ff(p)p 957 584 49 4 v 82 x Fn(2)o Fj(\024h)p Fn(\(1)h(+)f Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p 874 711 728 4 v 1102 802 a(2)p Fj(f)g Fn(\(0\))1335 773 y Fg(3)1611 588 y Fh(!)1694 734 y Fn(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)1174 b Fn(\(2.31\))-94 958 y(According)31 b(to)g(\(2.28\),)h(\(2.29\))f(and)h(\(2.30\),)f(w)m (e)i(get)e Ff(\000)p Fn(\()p Fj(A)2065 922 y Fe(0)2110 958 y Ff(\000)21 b Fj(y)t Fn(\))2298 922 y Fe(00)2360 958 y Fn(+)f Fj(f)2515 922 y Fg(2)2554 958 y Fj(A)2627 922 y Fe(0)2671 958 y Ff(\000)h Fj(f)11 b Fn(\(0\))2953 922 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b(w)m(e)g(ha)m(v)m(e)g(the)f (equalit)m(y)f(\(cf.)43 b(\(2.18\)\))-53 3447 y Ff(\000)p Fn(\()p Fj(z)27 b Ff(\000)c Fj(W)14 b Fn(\))378 3410 y Fe(00)442 3447 y Fn(+)22 b Fj(v)591 3410 y Fg(2)587 3471 y(2)630 3447 y Fn(\()p Fj(z)27 b Ff(\000)c Fj(W)14 b Fn(\))27 b(=)h Ff(\000)p Fj(z)t Fn(\(0\))17 b(exp)q(\()p Ff(\000)p Fj(\036)p Fn(\()p Fj(x)p Fn(\)\)\()p Ff(\000)2027 3407 y Fi(\024)p 1999 3423 95 4 v 1999 3433 a Fe(p)p 2059 3433 36 3 v 2059 3488 a Fg(2)2323 3407 y(1)p 2114 3423 453 4 v 2114 3484 a(cosh)2248 3459 y Fd(2)2282 3484 y Fg(\()2324 3457 y Fb(\024x)p 2320 3469 81 4 v 2320 3478 a Fa(p)p 2369 3478 31 3 v 2369 3520 a Fd(2)2410 3484 y Fg(+)p Fi(x)2505 3493 y Fd(0)2539 3484 y Fg(\))2599 3447 y Fn(+)22 b(2)p Fj(v)2793 3462 y Fg(2)2832 3447 y Fj(c)2874 3462 y Fg(1)2913 3447 y Fj(\024)h Fn(+)f Fj(c)3132 3410 y Fg(2)3132 3471 y(1)3171 3447 y Fj(\024)3227 3410 y Fg(2)3267 3447 y Fn(\))p Fj(:)3436 3469 y Fn(\(2.34\))-94 3645 y(W)-8 b(e)32 b(tak)m(e)i Fj(c)327 3660 y Fg(1)399 3645 y 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Fj(h)p Fn(\(1)22 b Ff(\000)h Fj(C)7 b(\024)1583 5322 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)1749 5363 y Fn(\))27 b Ff(\024)i(\000)p Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))p Fj(:)-94 5530 y Fn(F)-8 b(rom)31 b(\(2.35\),)g(it)h(results)h(that)-94 5687 y Fj(h)p Fn(\(1)22 b Ff(\000)g Fj(C)7 b(\024)303 5646 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)469 5687 y Fn(\))17 b(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))27 b Ff(\024)i Fj(A)1271 5646 y Fe(0)1294 5687 y Fj(:)2115 b Fn(\(2.36\))-94 5845 y(According)32 b(to)g(\(2.32\))g(and)h (\(2.36\),)e(the)i(pro)s(of)f(of)g(Prop)s(osition)f(2.5)h(follo)m(ws.)p eop %%Page: 8 8 8 7 bop 94 213 a Fn(8)94 511 y Fk(3.)57 b(Estimates)35 b(for)j Fj(A)1001 475 y Fe(0)1024 511 y Fn(\(0\))94 709 y Fk(3.1.)57 b(Preliminaries)192 830 y Fn(First,)48 b(w)m(e)e(sho)m(w)g (that)g(Inequalit)m(y)f(\(1.7\))g(is)f(true)i(for)f(some)g(pairs)g(\()p Fj(f)5 b(;)17 b(A)p Fn(\))45 b(suc)m(h)i(that)e Fj(f)11 b Fn(\(0\))49 b Ff(2)94 950 y Fn([0)p Fj(;)253 911 y Fg(1)p 224 927 95 4 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Fn(\(0\))27 b(=)3478 2578 y(1)p 3436 2622 132 4 v 3436 2640 a Ff(p)p 3519 2640 49 4 v 82 x Fn(2)3588 2645 y(+)10 b Fj(\013)q(\024)3803 2577 y Fd(1)p 3802 2589 31 4 v 3802 2630 a(4)3847 2645 y Fn(,)94 2838 y(w)m(e)34 b(get)f(for)f(some)816 2813 y(~)794 2838 y Fj(C)893 3096 y(\024)p Fn(\()p Fj(A)1060 3055 y Fe(0)1083 3096 y Fn(\(0\)\))1246 3055 y Fg(2)1313 3096 y Ff(\024)c Fn(2)1467 3055 y Fe(\000)1532 3027 y Fd(3)p 1532 3039 V 1532 3081 a(2)1598 3096 y Fn(+)1706 3028 y(1)p 1706 3072 49 4 v 1706 3164 a(2)1765 3096 y Fj(\036)1823 3055 y Fe(00)1823 3120 y Fg(0)1865 3096 y Fn(\()1955 3028 y(1)p 1913 3072 132 4 v 1913 3090 a Ff(p)p 1996 3090 49 4 v 82 x Fn(2)2055 3096 y(\))p Fj(\013)2156 3055 y Fg(2)2195 3096 y Fj(\024)2261 3027 y Fd(1)p 2261 3039 31 4 v 2261 3081 a(2)2328 3096 y Fn(+)2448 3071 y(~)2426 3096 y Fj(C)7 b(\024)2569 3027 y Fd(1)p 2569 3039 V 2569 3081 a(2)2635 3096 y Fn(+)22 b Ff(O)2812 3111 y Fi(\013)2862 3096 y Fn(\()p Fj(\024)2966 3027 y Fd(3)p 2966 3039 V 2966 3081 a(4)3011 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3964 132 4 v 927 3982 a Ff(p)p 1010 3982 49 4 v 82 x Fn(2)1091 3987 y(+)g Fj(\013)q(\024)1318 3919 y Fd(1)p 1318 3931 31 4 v 1318 3972 a(4)1362 3987 y Fn(])p Fj(;)45 b(\013)28 b(>)g Fn(0)p Fj(:)1942 b Fn(\(3.3\))94 4242 y(F)-8 b(rom)32 b(\(1.9\),)g(for)g(all)e(pairs)i(\()p Fj(f)5 b(;)17 b(A)p Fn(\))33 b(solution)e(of)h(\()p Fj(GL)p Fn(\))2112 4257 y Fe(1)2219 4242 y Fn(satisfying)g(\(3.3\),)g(w)m(e)i (ha)m(v)m(e)94 4458 y Fj(\024h)206 4417 y Fg(2)274 4458 y Ff(\025)28 b Fn(2)428 4417 y Fe(\000)493 4390 y Fd(3)p 493 4402 V 493 4443 a(2)559 4458 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)843 4390 y Fd(1)p 844 4402 V 844 4443 a(4)888 4458 y Fn(\))p Fj(:)2720 b Fn(\(3.4\))94 4661 y(F)-8 b(rom)32 b(\(1.11\))f(and)i(\(3.4\),)f(w)m(e)i(deduce)g(that)94 4876 y Fj(\024h)206 4835 y Fg(2)274 4876 y Fn(=)27 b(2)426 4835 y Fe(\000)491 4808 y Fd(3)p 491 4820 V 491 4861 a(2)558 4876 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)842 4808 y Fd(1)p 842 4820 V 842 4861 a(4)886 4876 y Fn(\))p Fj(;)2722 b Fn(\(3.5\))94 5079 y(for)33 b(all)d(pairs)i(\()p Fj(f)5 b(;)17 b(A)p Fn(\))33 b(solution)e(of)h(\()p Fj(GL)p Fn(\))1596 5094 y Fe(1)1703 5079 y Fn(satisfying)g(\(3.3\).)94 5199 y(In)38 b(order)g(to)f(get)h(Inequalit)m(y)f(\(1.7\),)i(let)e(us)h (analyze)f(the)h(pro)s(of)f(b)m(y)h(C.)g(Bolley)f(and)h(B.)f(Hel\013er) g(of)94 5320 y(the)h(De)f(Gennes)h(form)m(ula)d([5])i(and)g(impro)m(v)m (e)g(some)g(of)f(their)h(estimates.)57 b(F)-8 b(or)36 b(doing)g(this,)i(w)m(e)g(are)94 5440 y(guided)d(b)m(y)g(the)g (analysis)f(of)h(the)g(prop)s(erties)f(of)g(the)h(appro)m(ximate)f (solution)f(constructed)j(in)e([12].)94 5560 y(T)-8 b(o)33 b(get)g(an)f(upp)s(er)h(b)s(ound)g(on)g Fj(A)1325 5524 y Fe(0)1348 5560 y Fn(\(0\),)f(C.)h(Bolley)f(and)g(B.)h(Hel\013er)f (start)h(b)m(y)g(the)g(iden)m(tit)m(y)1411 5823 y Fj(h)1467 5782 y Fg(2)1534 5823 y Fn(=)28 b Ff(\000)p Fn(2)1781 5706 y Fh(Z)1864 5732 y Fg(+)p Fe(1)1827 5895 y Fg(0)2010 5823 y Fj(A)2083 5782 y Fe(0)2107 5823 y Fn(\()p Fj(t)p Fn(\))p Fj(A)2291 5782 y Fe(00)2333 5823 y Fn(\()p Fj(t)p Fn(\))p Fj(dt:)p eop %%Page: 9 9 9 8 bop 3636 213 a Fn(9)-94 511 y(Then,)33 b(from)f(\(1)p Fj(:)p Fn(2\))619 527 y Fg(\()p Fi(b)p Fg(\))707 511 y Fn(,)h(they)g(get)-94 752 y Fj(h)-38 711 y Fg(2)29 752 y Fn(=)27 b(2)198 635 y Fh(Z)281 661 y Fg(+)p Fe(1)244 823 y Fg(0)427 752 y Fj(A)500 711 y Fe(0)523 752 y Fn(\()p Fj(t)p Fn(\)\()p Ff(\000)p Fj(A)p Fn(\()p Fj(t)p Fn(\))p Fj(f)11 b Fn(\()p Fj(t)p Fn(\))1103 711 y Fg(2)1143 752 y Fn(\))p Fj(dt:)2191 b Fn(\(3.6\))-94 975 y(Then,)33 b(using)g(Inequalit)m(y)f(\(2.5\),)g(they)i(obtain)-94 1208 y Fj(h)-38 1167 y Fg(2)29 1208 y Ff(\024)28 b Fn(2)200 1091 y Fh(Z)282 1117 y Fg(+)p Fe(1)245 1280 y Fg(0)429 1208 y Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(A)672 1167 y Fe(0)695 1208 y Fn(\()p Fj(t)p Fn(\))806 1167 y Fg(2)845 1208 y Fj(dt:)2527 b Fn(\(3.7\))-94 1426 y(W)-8 b(e)32 b(susp)s(ect)j(that)d(something)g(has)h(b)s(een)g(lost)f(when)h (writing)e(:)1562 1630 y Ff(\000)p Fj(Af)39 b Ff(\024)29 b Fj(A)1978 1589 y Fe(0)2001 1630 y Fj(:)-94 1833 y Fn(Then,)k(using)g (\(1)p Fj(:)p Fn(2\))644 1849 y Fg(\()p Fi(a)p Fg(\))772 1833 y Fn(and)g(the)g(energy)g(conserv)-5 b(ation)33 b(\(2.4\),)f(they)i(get)-94 2074 y Fj(h)-38 2033 y Fg(2)29 2074 y Ff(\024)28 b Fn(2)200 1956 y Fh(Z)282 1983 y Fg(+)p Fe(1)245 2145 y Fg(0)429 2074 y Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(A)672 2033 y Fe(0)695 2074 y Fn(\()p Fj(t)p Fn(\))806 2033 y Fg(2)845 2074 y Fj(dt)28 b Fn(=)1062 1956 y Fh(Z)1145 1983 y Fg(+)p Fe(1)1109 2145 y Fg(0)1292 2074 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b Ff(\000)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1880 2033 y Fg(4)1920 2074 y Fn(\))p Fj(dx)22 b Ff(\000)g Fn(6)p Fj(\024)2290 2033 y Fe(\000)p Fg(2)2401 1956 y Fh(Z)2484 1983 y Fg(+)p Fe(1)2447 2145 y Fg(0)2631 2074 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)2880 2033 y Fe(0)2903 2074 y Fn(\()p Fj(x)p Fn(\))3034 2033 y Fg(2)3073 2074 y Fj(dx:)279 b Fn(\(3.8\))-94 2297 y(In)42 b(order)f(to)h(get)g(the)g(con)m(trol)f(of)g(the)h(righ)m (t-hand)f(side)h(of)f(Inequalit)m(y)h(\(3.8\),)i(in)d(particular)f(the) -94 2417 y(t)m(w)m(o)f(terms)f Fj(f)432 2381 y Fe(0)493 2417 y Fn(and)h(1)26 b Ff(\000)g Fj(f)926 2381 y Fg(2)965 2417 y Fn(,)40 b(they)f(use)h(the)f(conserv)-5 b(ation)38 b(la)m(w)g(\(2.4\).)60 b(If)38 b(w)m(e)i(rewrite)e(the)h(energy)-94 2538 y(conserv)-5 b(ation)32 b(in)g(the)h(form)334 2761 y(\()p Fj(\024)428 2720 y Fe(\000)p Fg(1)522 2761 y Fj(f)581 2720 y Fe(0)604 2761 y Fn(\()p Fj(x)p Fn(\))23 b(+)f Fj(A)929 2720 y Fe(0)952 2761 y Fn(\()p Fj(x)p Fn(\)\))1121 2720 y Fg(2)1188 2761 y Fn(=)28 b Fj(A)p Fn(\()p Fj(x)p Fn(\))1496 2720 y Fg(2)1536 2761 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1726 2720 y Fg(2)1787 2761 y Fn(+)22 b(2)p Fj(\024)1990 2720 y Fe(\000)p Fg(1)2085 2761 y Fj(f)2144 2720 y Fe(0)2167 2761 y Fn(\()p Fj(x)p Fn(\))p Fj(A)2371 2720 y Fe(0)2394 2761 y Fn(\()p Fj(x)p Fn(\))h(+)2656 2694 y(1)p 2656 2738 49 4 v 2656 2829 a(2)2714 2761 y(\(1)f Ff(\000)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))3113 2720 y Fg(2)3152 2761 y Fn(\))3190 2720 y Fg(2)3230 2761 y Fj(;)-94 3000 y Fn(w)m(e)22 b(observ)m(e)i(that)d(these)i(authors)f(ha)m(v)m(e)h (neglected)g(the)f(p)s(ositiv)m(e)f(term)g Fj(A)p Fn(\()p Fj(x)p Fn(\))2721 2964 y Fg(2)2761 3000 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))2951 2964 y Fg(2)2990 3000 y Fn(+2)p Fj(\024)3171 2964 y Fe(\000)p Fg(1)3266 3000 y Fj(f)3325 2964 y Fe(0)3348 3000 y Fn(\()p Fj(x)p Fn(\))p Fj(A)3552 2964 y Fe(0)3575 3000 y Fn(\()p Fj(x)p Fn(\))-94 3120 y(for)32 b(getting)f(the)i (inequalit)m(y:)1046 3362 y Fj(\024)1102 3321 y Fe(\000)p Fg(1)1196 3362 y Fj(f)1255 3321 y Fe(0)1278 3362 y Fn(\()p Fj(x)p Fn(\))23 b(+)f Fj(A)1603 3321 y Fe(0)1626 3362 y Fn(\()p Fj(x)p Fn(\))28 b Ff(\025)1942 3295 y Fn(1)p 1900 3339 132 4 v 1900 3357 a Ff(p)p 1983 3357 49 4 v 82 x Fn(2)2042 3362 y(\(1)22 b Ff(\000)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))2440 3321 y Fg(2)2480 3362 y Fn(\))p Fj(:)-94 3619 y Fn(T)-8 b(o)41 b(impro)m(v)m(e)g(Inequalit)m(y)g(\(3.8\),)j(in)c (the)i(next)g(sections,)j(w)m(e)d(get)f(an)h(upp)s(er)g(b)s(ound)f(and) h(a)f(lo)m(w)m(er)-94 3739 y(b)s(ound)36 b(for)f(the)i(di\013erence)g (\()p Fj(A)1085 3703 y Fe(0)1108 3739 y Fn(\))1146 3703 y Fg(2)1210 3739 y Ff(\000)25 b Fj(A)1385 3703 y Fg(2)1425 3739 y Fj(f)1484 3703 y Fg(2)1523 3739 y Fn(.)54 b(Moreo)m(v)m(er,)39 b(w)m(e)e(impro)m(v)m(e)e(the)i(con)m(trol)e(on)h Fj(f)3286 3703 y Fe(0)3345 3739 y Fn(giv)m(en)h(in)-94 3860 y(\(2.3\).)42 b(This)33 b(is)f(the)h(ob)5 b(ject)34 b(of)e(the)h(follo)m(wing)d (subsection.)-94 4058 y Fk(3.2.)56 b(Estimate)35 b(for)j Fj(f)841 4021 y Fe(0)3 4178 y Fn(In)k(order)g(to)g(get)f(an)h(estimate) f(on)h Fj(f)1447 4142 y Fe(0)1512 4178 y Fn(on)f(the)i(in)m(terv)-5 b(al)40 b([0)p Fj(;)17 b(\024)2371 4142 y Fe(\000)p Fi(\032)2466 4178 y Fn(],)44 b(w)m(e)f(establish)e(the)i(follo)m(wing)-94 4298 y(lemma:)-94 4488 y Fk(Lemma)37 b(3.2)49 b Fc(L)-5 b(et)36 b Fj(\032)29 b Ff(2)h Fn([0)p Fj(;)968 4448 y Fg(1)p 968 4464 36 4 v 968 4522 a(4)1014 4488 y Fn(])p Fc(.)47 b(Ther)-5 b(e)35 b(exists)h Fj(\024)1723 4503 y Fg(0)1798 4488 y Fc(such)f(that,)i(for)e(al)5 b(l)35 b Fj(\024)30 b Ff(\024)g Fj(\024)2796 4503 y Fg(0)2835 4488 y Fc(,)36 b(for)f(al)5 b(l)36 b(p)-5 b(airs)35 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))-94 4608 y Fc(solutions)34 b(of)h Fn(\()p Fj(GL)p Fn(\))648 4623 y Fe(1)757 4608 y Fc(satisfying)g(\(3.3\),)f(we)g(have)g(the)h(fol)5 b(lowing)34 b(estimate)-94 4857 y Fj(A)-21 4815 y Fe(0)2 4857 y Fj(f)61 4815 y Fe(0)112 4857 y Fn(=)265 4789 y Fj(\024)321 4753 y Fg(2)361 4789 y Fj(h)417 4753 y Fg(3)p 225 4833 272 4 v 225 4925 a Fn(2)p Fj(f)11 b Fn(\(0\))458 4896 y Fg(2)523 4857 y Fn(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))17 b(\(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2176 4815 y Fi(\032)2216 4857 y Fn(\)\))17 b Fj(;)99 b Ff(8)28 b Fj(x)g Ff(2)h Fn([0)p Fj(;)17 b(\024)2872 4815 y Fe(\000)p Fi(\032)2966 4857 y Fn(])p Fj(:)465 b Fn(\(3.9\))-94 5096 y Fk(Pro)s(of)-94 5217 y(Step)37 b(1:)50 b(Estimate)36 b(for)h Fj(A)1018 5180 y Fe(0)1041 5217 y Fj(f)1100 5180 y Fe(0)1161 5217 y Fk(at)g(the)h(p)s(oin)m(t)e Fj(x)28 b Fn(=)g Fj(\024)2026 5180 y Fe(\000)p Fi(\032)2121 5217 y Fj(:)-94 5337 y Fn(Let)34 b Fj(\032)e Ff(2)g Fn([0)p Fj(;)391 5298 y Fg(1)p 391 5314 36 4 v 391 5371 a(4)437 5337 y Fn(].)50 b(W)-8 b(e)35 b(set)g Fj(Y)53 b Fn(=)31 b Fj(A)1155 5301 y Fe(0)1179 5337 y Fj(f)1238 5301 y Fe(0)1261 5337 y Fn(.)50 b(As)35 b Fj(f)1543 5301 y Fe(0)1597 5337 y Ff(\025)d Fn(0)j(and)g Fj(A)2055 5301 y Fe(0)2109 5337 y Ff(\025)d Fn(0,)j(w)m(e)h(ha)m(v)m(e)g Fj(Y)53 b Ff(\025)32 b Fn(0.)50 b(In)35 b(order)g(to)f(use)-94 5457 y(Lemma)28 b(2.3)h(with)g Fj(d)f Fn(=)f Fj(\024)862 5421 y Fe(\000)p Fi(\032)958 5457 y Fn(,)j(w)m(e)h(estimate)d Fj(A)1616 5421 y Fe(0)1640 5457 y Fj(f)1699 5421 y Fe(0)1751 5457 y Fn(at)i(the)g(p)s(oin)m(t)f Fj(x)f Fn(=)f Fj(\024)2527 5421 y Fe(\000)p Fi(\032)2623 5457 y Fn(.)42 b(F)-8 b(rom)28 b(\(2.1\),)i(\(2.4\))f(at)h(the)-94 5578 y(p)s(oin)m(t)h Fj(x)d Fn(=)g Fj(\024)403 5542 y Fe(\000)p Fi(\032)531 5578 y Fn(and)k(\(2.5\),)h(w)m(e)g(deduce)h(that) 731 5804 y Fj(f)790 5763 y Fe(0)813 5804 y Fn(\()p Fj(\024)907 5763 y Fe(\000)p Fi(\032)1002 5804 y Fn(\))28 b Ff(\024)1221 5737 y Fj(\024)p 1183 5781 132 4 v 1183 5799 a Ff(p)p 1266 5799 49 4 v 82 x Fn(2)1325 5804 y(\(1)21 b Ff(\000)i Fj(f)11 b Fn(\()p Fj(\024)1686 5763 y Fe(\000)p Fi(\032)1781 5804 y Fn(\))1819 5763 y Fg(2)1858 5804 y Fn(\))28 b Ff(\024)2077 5737 y Fj(\024)p 2039 5781 132 4 v 2039 5799 a Ff(p)p 2122 5799 49 4 v 82 x Fn(2)2181 5804 y(\(1)22 b Ff(\000)g Fn(tanh)q(\()p Fj(x)2678 5819 y Fg(0)2717 5804 y Fn(\))2755 5763 y Fg(2)2795 5804 y Fn(\))p Fj(:)p eop %%Page: 10 10 10 9 bop 94 213 a Fn(10)94 511 y(According)29 b(to)e(\(2.27\),)i(and)f (as)h(tanh\()p Fj(x)1563 526 y Fg(0)1603 511 y Fn(\))e(=)h Fj(f)11 b Fn(\(0\))27 b(with)h Fj(f)11 b Fn(\(0\))27 b Ff(2)h Fn([)2573 472 y Fg(1)p 2543 488 95 4 v 2543 498 a Fe(p)p 2602 498 36 3 v 55 x Fg(2)2661 511 y Ff(\000)13 b Fj(\013)q(\024)2880 448 y Fd(1)p 2881 460 31 4 v 2881 501 a(4)2925 511 y Fj(;)3008 472 y Fg(1)p 2979 488 95 4 v 2979 498 a Fe(p)p 3038 498 36 3 v 55 x Fg(2)3096 511 y Fn(+)g Fj(\013)q(\024)3314 448 y Fd(1)p 3314 460 31 4 v 3314 501 a(4)3359 511 y Fn(],)29 b(w)m(e)g(deduce)94 664 y(that)k(there)g(exists)847 638 y(^)824 664 y Fj(C)40 b Fn(suc)m(h)34 b(that)94 904 y(\()p Fj(A)205 863 y Fe(0)229 904 y Fj(f)288 863 y Fe(0)311 904 y Fn(\)\()p Fj(\024)443 863 y Fe(\000)p Fi(\032)538 904 y Fn(\))28 b Ff(\024)709 758 y Fh( )819 837 y Fj(\024h)p 785 881 181 4 v 785 981 a Fn(2)834 899 y Ff(p)p 916 899 49 4 v 916 981 a Fn(2)997 904 y(+)1117 879 y(^)1095 904 y Fj(C)7 b(\024)1238 836 y Fd(3)p 1238 848 31 4 v 1238 889 a(4)1283 758 y Fh(!)1365 904 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)1869 863 y Fe(\000)p Fi(\032)1964 904 y Fn(\))p Fj(:)1596 b Fn(\(3.10\))94 1171 y(T)-8 b(o)41 b(get)g(a)f(lo)m(w)m(er)h(b)s(ound) g(for)f(\()p Fj(A)1346 1135 y Fe(0)1370 1171 y Fj(f)1429 1135 y Fe(0)1452 1171 y Fn(\)\()p Fj(\024)1584 1135 y Fe(\000)p Fi(\032)1679 1171 y Fn(\),)j(let)d(us)h(remark)f(that,)j(at)d (the)h(p)s(oin)m(t)f Fj(x)i Fn(=)g Fj(\024)3496 1135 y Fe(\000)p Fi(\032)3591 1171 y Fn(,)h(for)d Fj(\024)94 1292 y Fn(small,)31 b(w)m(e)i(ha)m(v)m(e)830 1512 y(\()p Fj(A)941 1471 y Fg(2)980 1512 y Fj(f)1039 1471 y Fg(2)1100 1512 y Ff(\000)23 b Fj(A)1273 1471 y Fe(0)1297 1460 y Fg(2)1336 1512 y Fn(\)\()p Fj(\024)1468 1471 y Fe(\000)p Fi(\032)1563 1512 y Fn(\))28 b(=)f Ff(O)s Fn(\()p Fj(\024)1908 1471 y Fe(\000)p Fg(1)2003 1512 y Fn(\))17 b(exp\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(\024)2561 1471 y Fe(\000)p Fi(\032)2657 1512 y Fn(\))27 b(=)60 b Ff(O)s Fn(\()p Fj(\024)3034 1471 y Fg(2)3074 1512 y Fn(\))p Fj(:)94 1732 y Fn(F)-8 b(rom)32 b(\(2.4\))g(at)g(the)h(p)s(oin)m(t)f Fj(x)c Fn(=)f Fj(\024)1368 1695 y Fe(\000)p Fi(\032)1464 1732 y Fn(,)32 b(w)m(e)i(deduce)g(that)340 2014 y Fj(\024)396 1973 y Fe(\000)p Fg(2)491 2014 y Fj(f)550 1973 y Fe(0)573 2014 y Fn(\()p Fj(\024)667 1973 y Fe(\000)p Fi(\032)762 2014 y Fn(\))800 1973 y Fg(2)867 2014 y Fn(=)981 1946 y(\(1)21 b Ff(\000)i Fj(f)11 b Fn(\()p Fj(\024)1342 1910 y Fe(\000)p Fi(\032)1437 1946 y Fn(\))1475 1910 y Fg(2)1514 1946 y Fn(\))1552 1910 y Fg(2)p 981 1990 612 4 v 1262 2082 a Fn(2)1624 2014 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1898 1973 y Fg(2)1937 2014 y Fn(\))28 b Ff(\025)2118 1946 y Fn(1)p 2118 1990 49 4 v 2118 2082 a(2)2194 1868 y Fh( )2259 2014 y Fn(1)22 b Ff(\000)h Fn(tanh)2625 1971 y Fg(2)2664 2014 y Fn(\()2712 1946 y Fj(\024)2768 1910 y Fg(1)p Fe(\000)p Fi(\032)p 2712 1990 187 4 v 2740 2008 a Ff(p)p 2823 2008 49 4 v 82 x Fn(2)2931 2014 y(+)f Fj(x)3084 2029 y Fg(0)3124 2014 y Fn(\))3162 1868 y Fh(!)3250 2014 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)3524 1973 y Fg(2)3563 2014 y Fn(\))p Fj(:)94 2296 y Fn(Hence,)34 b(from)e(\(3.3\),)g(w)m(e)i(get)1364 2539 y Fj(f)1423 2498 y Fe(0)1446 2539 y Fn(\()p Fj(\024)1540 2498 y Fe(\000)p Fi(\032)1636 2539 y Fn(\))27 b Ff(\025)1879 2471 y Fj(\024)p 1816 2516 181 4 v 1816 2615 a Fn(2)1865 2533 y Ff(p)p 1948 2533 49 4 v 82 x Fn(2)2024 2442 y Fh(\020)2073 2539 y Fn(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)2428 2471 y Fd(3)p 2428 2483 31 4 v 2428 2524 a(4)2473 2539 y Fn(\))2511 2442 y Fh(\021)2577 2539 y Fj(:)94 2828 y Fn(According)33 b(to)f(\(2.27\),)g(it)g(results)g(that)h(there)g (exists)2146 2803 y(~)2123 2828 y Fj(C)40 b Fn(suc)m(h)34 b(that)94 2948 y Fh( )204 3026 y Fj(\024h)p 170 3071 181 4 v 170 3170 a Fn(2)219 3088 y Ff(p)p 302 3088 49 4 v 82 x Fn(2)383 3094 y(+)503 3069 y(~)481 3094 y Fj(C)7 b(\024)624 3025 y Fd(3)p 624 3037 31 4 v 624 3079 a(4)668 2948 y Fh(!)751 3094 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)1255 3053 y Fe(\000)p Fi(\032)1350 3094 y Fn(\))28 b Ff(\024)g Fn(\()p Fj(A)1632 3053 y Fe(0)1655 3094 y Fj(f)1714 3053 y Fe(0)1737 3094 y Fn(\)\()p Fj(\024)1869 3053 y Fe(\000)p Fi(\032)1964 3094 y Fn(\))1623 b(\(3.11\))94 3359 y(Finally)-8 b(,)30 b(from)i(\(3.10\))g(and)g (\(3.11\),)g(w)m(e)i(get)94 3479 y Fh( )204 3557 y Fj(\024h)p 170 3602 181 4 v 170 3702 a Fn(2)219 3620 y Ff(p)p 302 3620 49 4 v 82 x Fn(2)383 3625 y(+)503 3600 y(~)481 3625 y Fj(C)7 b(\024)624 3557 y Fd(3)p 624 3569 31 4 v 624 3610 a(4)668 3625 y Fn(\))706 3479 y Fh(!)789 3625 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)1293 3584 y Fe(\000)p Fi(\032)1388 3625 y Fn(\))28 b Ff(\024)g Fn(\()p Fj(A)1670 3584 y Fe(0)1693 3625 y Fj(f)1752 3584 y Fe(0)1775 3625 y Fn(\)\()p Fj(\024)1907 3584 y Fe(\000)p Fi(\032)2002 3625 y Fn(\))g Ff(\024)2173 3479 y Fh( )2283 3557 y Fj(\024h)p 2249 3602 181 4 v 2249 3702 a Fn(2)2298 3620 y Ff(p)p 2381 3620 49 4 v 82 x Fn(2)2461 3625 y(+)2582 3600 y(^)2559 3625 y Fj(C)7 b(\024)2702 3557 y Fd(3)p 2702 3569 31 4 v 2702 3610 a(4)2747 3479 y Fh(!)2829 3625 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)3333 3584 y Fe(\000)p Fi(\032)3428 3625 y Fn(\))p Fj(:)132 b Fn(\(3.12\))94 3885 y Fk(Step)38 b(2:)50 b(Upp)s(er)38 b(b)s(ound)h(for)e(Y.)94 4005 y Fn(W)-8 b(e)33 b(ha)m(v)m(e)1597 4126 y Fj(Y)1675 4085 y Fe(0)1726 4126 y Fn(=)27 b Fj(A)1902 4085 y Fe(00)1945 4126 y Fj(f)2004 4085 y Fe(0)2049 4126 y Fn(+)22 b Fj(A)2220 4085 y Fe(0)2244 4126 y Fj(f)2303 4085 y Fe(00)2345 4126 y Fj(:)94 4300 y Fn(Using)33 b(the)g (Ginzburg-Landau)e(equations)i(\(1.2\),)f(w)m(e)i(get)1211 4520 y Fj(Y)1289 4479 y Fe(0)1340 4520 y Fn(=)27 b Fj(Af)1575 4479 y Fg(2)1615 4520 y Fj(f)1674 4479 y Fe(0)1719 4520 y Fn(+)22 b Fj(\024)1873 4479 y Fg(2)1912 4520 y Fj(A)1985 4479 y Fe(0)2009 4520 y Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)2362 4479 y Fg(3)2424 4520 y Fn(+)g Fj(A)2595 4479 y Fg(2)2634 4520 y Fj(f)11 b Fn(\))p Fj(:)94 4740 y Fn(W)-8 b(e)38 b(observ)m(e)g(that)f Fj(Y)912 4704 y Fe(0)935 4740 y Fn(\(0\))e(=)g Fj(h)17 b(f)1338 4704 y Fe(00)1379 4740 y Fn(\(0\).)56 b(Di\013eren)m(tiating)34 b(once)k(more)e(and)h(using)f(\(1.2\),)i(w)m(e)f(deduce)94 4860 y(that)c(the)g(function)f Fj(Y)54 b Fn(is)32 b(the)h(unique)g (solution)e(in)h Fj(H)2122 4824 y Fg(2)2161 4860 y Fn(\()p Fj(I)-25 b(R)2292 4824 y Fg(+)2350 4860 y Fn(\))33 b(of)f(the)h (problem)127 5002 y Fh(8)127 5076 y(>)127 5101 y(<)127 5251 y(>)127 5276 y(:)242 5077 y Fj(Y)321 5041 y Fe(00)385 5077 y Ff(\000)23 b Fj(f)544 5041 y Fg(2)583 5077 y Fj(Y)242 5203 y Fn(=)28 b(2)p Fj(f)11 b(f)513 5167 y Fe(0)535 5156 y Fg(2)575 5203 y Fj(A)22 b Fn(+)g(2)p Fj(\024)873 5167 y Fg(2)913 5203 y Fj(Af)1045 5167 y Fg(2)1084 5203 y Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)1437 5167 y Fg(3)1498 5203 y Fn(+)g Fj(A)1669 5167 y Fg(2)1709 5203 y Fj(f)11 b Fn(\))22 b(+)g Fj(\024)1982 5167 y Fg(2)2021 5203 y Fj(A)2094 5167 y Fe(0)2118 5203 y Fn(\()p Ff(\000)p Fj(f)2292 5167 y Fe(0)2337 5203 y Fn(+)g(3)p Fj(f)2543 5167 y Fe(0)2566 5203 y Fj(f)2625 5167 y Fg(2)2664 5203 y Fn(\))g(+)g(2)p Fj(\024)2927 5167 y Fg(2)2967 5203 y Fj(A)3040 5167 y Fe(0)3063 5156 y Fg(2)3103 5203 y Fj(Af)33 b Fn(+)22 b(2)p Fj(\024)3460 5167 y Fg(2)3499 5203 y Fj(A)3572 5167 y Fe(0)3595 5203 y Fj(A)3668 5167 y Fg(2)3708 5203 y Fj(f)3767 5167 y Fe(0)242 5323 y Fj(Y)321 5287 y Fe(0)344 5323 y Fn(\(0\))27 b(=)h Fj(h)17 b(f)732 5287 y Fe(00)774 5323 y Fn(\(0\))p Fj(:)94 5535 y Fn(Hence,)34 b(w)m(e)g(get)94 5649 y Fh(\()203 5737 y Ff(\000)p Fj(Y)359 5701 y Fe(00)423 5737 y Fn(+)22 b Fj(G)17 b(Y)49 b Fn(=)27 b Ff(\000)p Fn(2)p Fj(f)11 b(f)1068 5701 y Fe(0)1091 5690 y Fg(2)1131 5737 y Fj(A)22 b Ff(\000)h Fn(2)p Fj(\024)1431 5701 y Fg(2)1470 5737 y Fj(Af)1602 5701 y Fg(2)1641 5737 y Fn(\()p Ff(\000)p Fj(f)34 b Fn(+)22 b Fj(f)1995 5701 y Fg(3)2034 5737 y Fn(\))g Ff(\000)g Fn(2)p Fj(\024)2298 5701 y Fg(2)2338 5737 y Fj(A)2411 5701 y Fg(3)2450 5737 y Fj(f)2509 5701 y Fg(3)2571 5737 y Ff(\000)g Fn(2)p Fj(\024)2775 5701 y Fg(2)2815 5737 y Fj(A)2888 5701 y Fe(0)2911 5690 y Fg(2)2950 5737 y Fj(Af)5 b(;)203 5857 y(Y)281 5821 y Fe(0)304 5857 y Fn(\(0\))28 b(=)f Fj(h)17 b(f)692 5821 y Fe(00)734 5857 y Fn(\(0\))p Fj(;)3625 5795 y Fn(\(3.13\))p eop %%Page: 11 11 11 10 bop 3587 213 a Fn(11)-94 511 y(where)-94 691 y Fj(G)27 b Fn(:=)h Fj(f)200 650 y Fg(2)261 691 y Fn(+)22 b Fj(\024)415 650 y Fg(2)455 691 y Fn(\()p Ff(\000)p Fn(1)g(+)g(3)p Fj(f)847 650 y Fg(2)908 691 y Fn(+)g Fj(A)1079 650 y Fg(2)1119 691 y Fn(\))p Fj(:)2252 b Fn(\(3.14\))-94 871 y(According)32 b(to)g(\(2.6\))g(and)h(\(2.9\),)f(for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1673 835 y Fe(\000)p Fi(\032)1768 871 y Fn(],)33 b(w)m(e)g(get)-94 1051 y Fj(f)11 b Fn(\(0\))90 1010 y Fg(2)156 1051 y Ff(\024)28 b Fj(G)p Fn(\()p Fj(x)p Fn(\))g Ff(\024)g Fj(f)11 b Fn(\(0\))786 1010 y Fg(2)847 1051 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)1121 1010 y Fg(1)p Fe(\000)p Fi(\032)1252 1051 y Fn(\))p Fj(:)2119 b Fn(\(3.15\))-94 1231 y(In)31 b(order)h(to)f(get)g(an)g(upp)s(er)h(b)s (ound)f(for)g Fj(Y)22 b Fn(,)31 b(w)m(e)i(lo)s(ok)d(for)g(an)i(upp)s (er)f(b)s(ound)h(of)f(the)h(righ)m(t-hand)e(side)-94 1351 y(of)i(\(3.13\).)42 b(According)33 b(to)f(\(2.1\),)g(\(2.2\))g (and)h(\(2.3\),)f(w)m(e)i(get)979 1545 y(2)p Fj(f)11 b Fn(\()p Fj(f)1184 1509 y Fe(0)1207 1545 y Fn(\))1245 1509 y Fg(2)1284 1545 y Fj(A)28 b Ff(\024)g Fn(2)p Fj(f)11 b(f)1657 1509 y Fe(0)1680 1498 y Fg(2)1719 1545 y Fj(A)22 b Ff(\000)h Fn(2)p Fj(\024)2019 1509 y Fg(2)2058 1545 y Fj(Af)2190 1509 y Fg(2)2230 1545 y Fn(\()p Fj(f)32 b Ff(\000)23 b Fj(f)2507 1509 y Fg(3)2546 1545 y Fn(\))p Fj(:)-94 1758 y Fn(As)36 b Fj(h)e Fn(=)g Ff(O)s Fn(\()p Fj(\024)429 1721 y Fe(\000)494 1694 y Fd(1)p 494 1706 31 4 v 494 1747 a(2)539 1758 y Fn(\))i(\(see)h(\(1.11\)\),)f(and)h (from)e(\(2.3\),)h(\(2.9\),\(2.27\))f(and)i(\(3.3\),)f(there)h(exists)g Fj(C)3422 1773 y Fg(1)3498 1758 y Fn(suc)m(h)-94 1878 y(that,)32 b(for)g(all)e Fj(x)f Ff(2)f Fn([0)p Fj(;)17 b(\024)782 1842 y Fe(\000)p Fi(\032)877 1878 y Fn(],)32 b(w)m(e)i(get)1137 2084 y(2)p Fj(f)11 b Fn(\()p Fj(f)1342 2043 y Fe(0)1365 2084 y Fn(\))1403 2043 y Fg(2)1442 2084 y Fj(A)28 b Ff(\025)g Fj(C)1718 2099 y Fg(1)1757 2084 y Fj(\024)1823 2015 y Fd(3)p 1824 2027 V 1824 2069 a(2)1885 2084 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 2276 y Fn(F)-8 b(rom)31 b(\(3.13\),)g(w)m(e)j(deduce)g (that,)f(for)f Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1680 2240 y Fe(\000)p Fi(\032)1775 2276 y Fn(],)33 b(w)m(e)g(ha)m(v)m(e)581 2474 y Ff(\000)p Fj(Y)736 2437 y Fe(00)801 2474 y Fn(+)22 b Fj(GY)49 b Ff(\024)28 b(\000)p Fn(2)p Fj(\024)1369 2437 y Fg(2)1409 2474 y Fj(Af)11 b Fn(\()p Fj(A)1652 2437 y Fg(2)1691 2474 y Fj(f)1750 2437 y Fg(2)1812 2474 y Fn(+)22 b Fj(A)1983 2437 y Fe(0)2006 2427 y Fg(2)2045 2474 y Fn(\))g Ff(\000)h Fj(C)2275 2489 y Fg(1)2314 2474 y Fj(\024)2380 2410 y Fd(3)p 2380 2422 V 2380 2463 a(2)2442 2474 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 2680 y Fn(No)m(w,)36 b(let)f(us)h(analyze)f(the)g (main)f(term)h Ff(\000)p Fn(2)p Fj(\024)1617 2643 y Fg(2)1657 2680 y Fj(Af)11 b Fn(\()p Fj(A)1900 2643 y Fg(2)1939 2680 y Fj(f)1998 2643 y Fg(2)2061 2680 y Fn(+)24 b Fj(A)2234 2643 y Fe(0)2257 2633 y Fg(2)2297 2680 y Fn(\).)51 b(According)35 b(to)g(\(2.6\),)g(\(2.9\),)h(and)-94 2800 y(Prop)s(osition)30 b(2.5,)j(w)m(e)g(get)-53 2989 y Ff(\000)p Fn(2)p Fj(\024)129 2953 y Fg(2)169 2989 y Fj(Af)11 b Fn(\()p Fj(A)412 2953 y Fg(2)451 2989 y Fj(f)510 2953 y Fg(2)572 2989 y Fn(+)22 b Fj(A)743 2953 y Fe(0)766 2943 y Fg(2)805 2989 y Fn(\))-53 3175 y Ff(\024)52 3029 y Fh( )118 3175 y Fn(2)p Fj(\024)223 3139 y Fg(2)336 3107 y Fj(h)392 3071 y Fg(3)p 272 3152 223 4 v 272 3243 a Fj(f)11 b Fn(\(0\))456 3214 y Fg(3)505 3175 y Fn(\()p Fj(f)g Fn(\(0\))21 b(+)h Ff(O)s Fn(\()p Fj(\024)1022 3134 y Fg(1)p Fe(\000)p Fi(\032)1153 3175 y Fn(\)\))1229 3134 y Fg(3)1291 3175 y Fn(+)g(2)p Fj(\024)1494 3134 y Fg(2)1606 3107 y Fj(h)p 1543 3152 184 4 v 1543 3243 a(f)11 b Fn(\(0\))1753 3079 y Fh(\020)1802 3175 y Fj(f)g Fn(\(0\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2282 3134 y Fg(1)p Fe(\000)p Fi(\032)2413 3175 y Fn(\))2451 3079 y Fh(\021)2517 3175 y Fn(\()p Fj(h)g Fn(+)g Ff(O)s Fn(\()p Fj(\024)2917 3107 y Fd(1)p 2917 3119 31 4 v 2917 3160 a(2)2957 3134 y Fe(\000)p Fg(2)p Fi(\032)3088 3175 y Fn(\)\))3164 3134 y Fg(2)3203 3029 y Fh(!)3286 3175 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 3431 y Fn(It)32 b(results)h(that)g(there)g (exists)g Fj(C)1121 3446 y Fg(2)1193 3431 y Fn(suc)m(h)h(that,)e(for)g Fj(x)d Ff(2)f Fn([0)p Fj(;)17 b(\024)2154 3395 y Fe(\000)p Fi(\032)2249 3431 y Fn(],)32 b(w)m(e)i(get)586 3636 y Ff(\000)p Fn(2)p Fj(\024)768 3595 y Fg(2)808 3636 y Fj(Af)11 b Fn(\()p Fj(A)1051 3595 y Fg(2)1091 3636 y Fj(f)1150 3595 y Fg(2)1211 3636 y Fn(+)22 b Fj(A)1382 3595 y Fe(0)1405 3584 y Fg(2)1445 3636 y Fn(\))27 b Ff(\024)1615 3540 y Fh(\020)1665 3636 y Fn(4)p Fj(\024)1770 3595 y Fg(2)1809 3636 y Fj(h)1865 3595 y Fg(3)1927 3636 y Fn(+)22 b Fj(C)2095 3651 y Fg(2)2134 3636 y Fj(\024)2190 3595 y Fg(1)p Fe(\000)p Fi(\032)2321 3540 y Fh(\021)2387 3636 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 3828 y Fn(Then,)55 b(to)49 b(get)h(an)g(upp)s(er)g(b)s(ound)g (for)f Fj(Y)22 b Fn(,)54 b(w)m(e)d(compare)e(this)h(function)f(with)g (the)h(solution)f(in)-94 3949 y Fj(H)-5 3913 y Fg(2)33 3949 y Fn(\([0)p Fj(;)17 b(\024)247 3913 y Fe(\000)p Fi(\032)342 3949 y Fn(]\))33 b(of)-94 4142 y Ff(\000)p Fj(y)31 4157 y Fg(1)70 4101 y Fe(00)135 4142 y Fn(+)22 b Fj(f)11 b Fn(\(0\))417 4101 y Fg(2)455 4142 y Fj(y)503 4157 y Fg(1)570 4142 y Fn(=)674 4045 y Fh(\020)723 4142 y Fn(4)p Fj(\024)828 4101 y Fg(2)868 4142 y Fj(h)924 4101 y Fg(3)985 4142 y Fn(+)22 b Fj(C)1153 4157 y Fg(2)1193 4142 y Fj(\024)1249 4101 y Fg(1)p Fe(\000)p Fi(\032)1379 4045 y Fh(\021)1445 4142 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b Ff(\000)h Fj(C)2227 4157 y Fg(1)2266 4142 y Fj(\024)2332 4073 y Fd(3)p 2332 4085 V 2332 4127 a(2)2394 4142 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)474 b Fn(\(3.16\))-94 4322 y(satisfying)-94 4558 y Fj(y)-42 4517 y Fe(0)-46 4583 y Fg(1)-7 4558 y Fn(\(0\))27 b(=)g Fj(Y)327 4517 y Fe(0)350 4558 y Fn(\(0\))p Fj(;)72 b(y)622 4573 y Fg(1)661 4558 y Fn(\()p Fj(\024)755 4517 y Fe(\000)p Fi(\032)850 4558 y Fn(\))28 b(=)1019 4412 y Fh( )1129 4491 y Fj(\024h)p 1095 4535 181 4 v 1095 4635 a Fn(2)1144 4553 y Ff(p)p 1227 4553 49 4 v 82 x Fn(2)1307 4558 y(+)1428 4533 y(^)1405 4558 y Fj(C)7 b(\024)1548 4490 y Fd(3)p 1549 4502 31 4 v 1549 4543 a(4)1593 4412 y Fh(!)1675 4558 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)2179 4517 y Fe(\000)p Fi(\032)2275 4558 y Fn(\))p Fj(:)1096 b Fn(\(3.17\))-94 4800 y(F)-8 b(rom)31 b(\(3.15\))g(and)i(as)g Fj(Y)49 b Ff(\025)28 b Fn(0,)k(w)m(e)i(ha)m(v)m(e)435 4992 y Ff(\000)p Fn(\()p Fj(Y)44 b Ff(\000)23 b Fj(y)799 5007 y Fg(1)838 4992 y Fn(\))876 4951 y Fe(00)940 4992 y Fn(+)f Fj(f)11 b Fn(\(0\))1222 4951 y Fg(2)1261 4992 y Fn(\()p Fj(Y)43 b Ff(\000)23 b Fj(y)1547 5007 y Fg(1)1586 4992 y Fn(\))28 b Ff(\024)g(\000)p Fn(\()p Fj(Y)44 b Ff(\000)22 b Fj(y)2120 5007 y Fg(1)2159 4992 y Fn(\))2197 4951 y Fe(00)2262 4992 y Fn(+)g Fj(GY)43 b Ff(\000)23 b Fj(f)11 b Fn(\(0\))2821 4951 y Fg(2)2860 4992 y Fj(y)2908 5007 y Fg(1)2974 4992 y Ff(\024)28 b Fn(0)p Fj(:)-94 5185 y Fn(F)-8 b(rom)24 b(\(3.12\))h(and)g(\(3.17\),)i(w)m(e)g(ha)m(v)m(e)g (\()p Fj(Y)i Ff(\000)8 b Fj(y)1528 5200 y Fg(1)1567 5185 y Fn(\))1605 5149 y Fe(0)1629 5185 y Fn(\(0\))27 b(=)h(0)d(and)h(\()p Fj(Y)j Ff(\000)8 b Fj(y)2399 5200 y Fg(1)2439 5185 y Fn(\)\()p Fj(\024)2571 5149 y Fe(\000)p Fi(\032)2666 5185 y Fn(\))27 b Ff(\024)i Fn(0.)41 b(Applying)24 b(Lemma)-94 5305 y(2.3)33 b(with)g Fj(d)d Fn(=)g Fj(\024)530 5269 y Fe(\000)p Fi(\032)625 5305 y Fn(,)k Fj(C)j Fn(=)30 b Fj(f)11 b Fn(\(0\))1083 5269 y Fg(2)1155 5305 y Fn(and)34 b Fj(u)c Fn(=)f Fj(Y)45 b Ff(\000)23 b Fj(y)1787 5320 y Fg(1)1826 5305 y Fn(,)34 b(for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)2395 5269 y Fe(\000)p Fi(\032)2490 5305 y Fn(],)34 b(w)m(e)h(get)f Fj(Y)51 b Ff(\024)30 b Fj(y)3150 5320 y Fg(1)3223 5305 y Fn(on)k([0)p Fj(;)17 b(\024)3536 5269 y Fe(\000)p Fi(\032)3631 5305 y Fn(].)-94 5426 y(The)33 b(solution)e(of)h(\(3.16\))g(is)g(giv)m(en)h(b)m(y)-53 5610 y Fj(y)-5 5625 y Fg(1)34 5610 y Fn(\()p Fj(x)p Fn(\))28 b(=)319 5584 y(~)297 5610 y Fj(C)367 5625 y Fg(1)406 5610 y Fn(\()p Fj(\024)p Fn(\))17 b(exp\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))23 b(+)1238 5584 y(~)1216 5610 y Fj(C)1286 5625 y Fg(2)1325 5610 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)q(\()p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))-53 5795 y Ff(\000)146 5728 y Fn(1)p 34 5772 272 4 v 34 5863 a(8)p Fj(f)g Fn(\(0\))267 5835 y Fg(2)333 5795 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))940 5699 y Fh(\020)989 5795 y Fn(4)p Fj(\024)1094 5754 y Fg(2)1133 5795 y Fj(h)1189 5754 y Fg(3)1251 5795 y Fn(+)22 b Fj(C)1419 5810 y Fg(2)1458 5795 y Fj(\024)1514 5754 y Fg(1)p Fe(\000)p Fi(\032)1645 5795 y Fn(\))1683 5699 y Fh(\021)1754 5795 y Fn(+)g(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))2410 5649 y Fh( )2476 5795 y Ff(\000)2645 5728 y Fj(C)2715 5743 y Fg(1)p 2563 5772 V 2563 5863 a Fn(4)p Fj(f)g Fn(\(0\))2796 5835 y Fg(2)2845 5795 y Fj(\024)2911 5727 y Fd(3)p 2911 5739 31 4 v 2911 5780 a(2)2956 5795 y Fn(\(1)21 b(+)h(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))3488 5649 y Fh(!)3570 5795 y Fj(:)3639 5731 y Fn(\(3.18\))p eop %%Page: 12 12 12 11 bop 94 213 a Fn(12)94 511 y(F)-8 b(or)31 b Fj(x)d Ff(2)g Fn([0)p Fj(;)17 b(\024)621 475 y Fe(\000)p Fi(\032)716 511 y Fn(],)32 b(\()p Fj(\032)c Ff(\024)1033 472 y Fg(1)p 1033 488 36 4 v 1033 546 a(4)1078 511 y Fn(\),)k(w)m(e)g(ha)m(v)m(e)h Fj(\024)1607 448 y Fd(3)p 1607 460 31 4 v 1607 501 a(2)1652 511 y Fn(\(1)19 b(+)h(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))27 b(=)h Ff(O)s Fn(\()p Fj(\024)p Fn(\))p Fj(:)j Fn(Hence,)i(from)d(\(3.17\))h(and)h(\(3.18\),)94 632 y(w)m(e)i(get)136 845 y Fj(y)184 860 y Fg(1)223 845 y Fn(\()p Fj(\024)317 809 y Fe(\000)p Fi(\032)412 845 y Fn(\))28 b(=)604 820 y(~)581 845 y Fj(C)651 860 y Fg(2)691 845 y Fn(\()p Fj(\024)p Fn(\))17 b(exp\()p Fj(f)11 b Fn(\(0\))p Fj(\024)1266 809 y Fe(\000)p Fi(\032)1361 845 y Fn(\))22 b(+)1519 749 y Fh(\020)1591 820 y Fn(~)1569 845 y Fj(C)1639 860 y Fg(1)1678 845 y Fn(\()p Fj(\024)p Fn(\))g(+)g Ff(O)s Fn(\()p Fj(\024)p Fn(\))2144 749 y Fh(\021)2210 845 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(\024)2714 809 y Fe(\000)p Fi(\032)2809 845 y Fn(\))136 994 y(=)240 898 y Fh(\020)323 955 y Fi(\024h)p 299 971 130 4 v 299 1036 a Fg(2)334 980 y Fe(p)p 393 980 36 3 v 56 x Fg(2)461 994 y Fn(+)581 969 y(^)559 994 y Fj(C)c(\024)702 930 y Fd(3)p 702 942 31 4 v 702 984 a(4)746 898 y Fh(\021)812 994 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)1316 958 y Fe(\000)p Fi(\032)1412 994 y Fn(\))p Fj(:)3625 920 y Fn(\(3.19\))94 1234 y(As)34 b Fj(\024h)28 b Fn(=)f Ff(O)s Fn(\()p Fj(\024)668 1170 y Fd(1)p 668 1182 V 668 1223 a(2)713 1234 y Fn(\),)32 b(from)g(\(3.19\),)g(it)f(results)i(that)117 1428 y(~)94 1454 y Fj(C)164 1469 y Fg(2)204 1454 y Fn(\()p Fj(\024)p Fn(\))27 b(=)467 1357 y Fh(\020)517 1454 y Ff(\000)616 1428 y Fn(~)594 1454 y Fj(C)664 1469 y Fg(1)703 1454 y Fn(\()p Fj(\024)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)1141 1385 y Fd(1)p 1142 1397 V 1142 1438 a(2)1186 1454 y Fn(\))1224 1357 y Fh(\021)1290 1454 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(\024)1843 1412 y Fe(\000)p Fi(\032)1938 1454 y Fn(\))p Fj(:)1622 b Fn(\(3.20\))94 1672 y(F)-8 b(rom)32 b(\(3.18\),)f(w)m(e)j(ha)m(v)m(e)518 1939 y Fj(y)570 1897 y Fe(0)566 1963 y Fg(1)605 1939 y Fn(\(0\))27 b(=)h Fj(f)11 b Fn(\(0\))1066 1913 y(~)1045 1939 y Fj(C)1115 1954 y Fg(2)1153 1939 y Fn(\()p Fj(\024)p Fn(\))22 b Ff(\000)h Fj(f)11 b Fn(\(0\))1613 1913 y(~)1591 1939 y Fj(C)1661 1954 y Fg(1)1700 1939 y Fn(\()p Fj(\024)p Fn(\))22 b(+)2053 1871 y(1)p 1962 1915 233 4 v 1962 2007 a(8)p Fj(f)11 b Fn(\(0\))2204 1939 y(\(12)p Fj(\024)2396 1897 y Fg(2)2435 1939 y Fj(h)2491 1897 y Fg(3)2553 1939 y Ff(\000)22 b Fn(2)p Fj(C)2771 1954 y Fg(1)2810 1939 y Fj(\024)2876 1870 y Fd(3)p 2876 1882 31 4 v 2876 1923 a(2)2943 1939 y Fn(+)g(3)p Fj(C)3160 1954 y Fg(2)3199 1939 y Fj(\024)3255 1897 y Fg(1)p Fe(\000)p Fi(\032)3386 1939 y Fn(\))p Fj(:)94 2216 y Fn(F)-8 b(rom)32 b(\(3.20\))f(and)i(the)g (condition)e Fj(y)1470 2180 y Fe(0)1466 2241 y Fg(1)1505 2216 y Fn(\(0\))c(=)h Fj(Y)1839 2180 y Fe(0)1863 2216 y Fn(\(0\))f(=)g Fj(h)17 b(f)2250 2180 y Fe(00)2292 2216 y Fn(\(0\),)33 b(it)e(results)i(that)1169 2472 y(~)1147 2498 y Fj(C)1217 2513 y Fg(1)1256 2498 y Fn(\()p Fj(\024)p Fn(\))28 b(=)f Ff(\000)1606 2430 y Fj(hf)1721 2394 y Fe(00)1764 2430 y Fn(\(0\))p 1606 2474 282 4 v 1656 2566 a Fj(f)11 b Fn(\(0\))1921 2498 y(+)2029 2430 y(3)p 2029 2474 49 4 v 2029 2566 a(2)2113 2430 y Fj(\024)2169 2394 y Fg(2)2209 2430 y Fj(h)2265 2394 y Fg(3)p 2097 2474 223 4 v 2097 2566 a Fj(f)g Fn(\(0\))2281 2537 y Fg(2)2352 2498 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)2626 2457 y Fg(1)p Fe(\000)p Fi(\032)2757 2498 y Fn(\))p Fj(:)94 2776 y Fn(F)-8 b(rom)32 b(\(1.2\),)g(\(2.9\))g(and)g(\(2.10\),)g(w)m(e)i(get) 94 3041 y Fj(f)153 3000 y Fe(00)196 3041 y Fn(\(0\))27 b(=)461 2973 y Fj(\024)517 2937 y Fg(2)557 2973 y Fj(h)613 2937 y Fg(2)p 461 3018 192 4 v 465 3109 a Fj(f)11 b Fn(\(0\))685 3041 y(+)22 b Ff(O)s Fn(\()p Fj(\024)959 3000 y Fg(2)998 3041 y Fn(\))p Fj(:)2562 b Fn(\(3.21\))94 3296 y(Hence,)34 b(w)m(e)g(get)117 3531 y(~)94 3556 y Fj(C)164 3571 y Fg(1)204 3556 y Fn(\()p Fj(\024)p Fn(\))27 b(=)517 3489 y Fj(\024)573 3452 y Fg(2)613 3489 y Fj(h)669 3452 y Fg(3)p 477 3533 272 4 v 477 3624 a Fn(2)p Fj(f)11 b Fn(\(0\))710 3596 y Fg(2)781 3556 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)1055 3515 y Fg(1)p Fe(\000)p Fi(\032)1185 3556 y Fn(\))p Fj(:)2375 b Fn(\(3.22\))94 3819 y(According)33 b(to)f(\(3.18\))g(and)h(\(3.20\),) e(for)h Fj(x)d Ff(2)f Fn([0)p Fj(;)17 b(\024)1960 3783 y Fe(\000)p Fi(\032)2055 3819 y Fn(],)32 b(w)m(e)i(get)136 4085 y Fj(y)184 4100 y Fg(1)223 4085 y Fn(\()p Fj(x)p Fn(\))28 b(=)g Ff(\000)613 4017 y Fj(\024)669 3981 y Fg(2)709 4017 y Fj(h)765 3981 y Fg(3)p 573 4062 V 573 4153 a Fn(2)p Fj(f)11 b Fn(\(0\))806 4124 y Fg(2)871 4085 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))136 4278 y(+)17 b(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\)[)819 4252 y(~)797 4278 y Fj(C)867 4293 y Fg(1)906 4278 y Fn(\()p Fj(\024)p Fn(\))22 b(+)1158 4181 y Fh(\020)1208 4278 y Ff(\000)1307 4252 y Fn(~)1285 4278 y Fj(C)1355 4293 y Fg(1)1394 4278 y Fn(\()p Fj(\024)p Fn(\))h(+)f Ff(O)s Fn(\()p Fj(\024)1833 4214 y Fd(1)p 1833 4226 31 4 v 1833 4267 a(2)1877 4278 y Fn(\))1915 4181 y Fh(\021)1981 4278 y Fn(exp)c(\(2)p Fj(f)11 b Fn(\(0\)\()p Fj(x)22 b Ff(\000)g Fj(\024)2688 4241 y Fe(\000)p Fi(\032)2784 4278 y Fn(\)\)])p Fj(:)3625 4158 y Fn(\(3.23\))94 4518 y(F)-8 b(rom)31 b(\(3.22\),)g(w)m(e)h(ha)m(v)m(e)1046 4492 y(~)1024 4518 y Fj(C)1094 4533 y Fg(1)1133 4518 y Fn(\()p Fj(\024)p Fn(\))c(=)g Ff(O)s Fn(\()p Fj(\024)1583 4454 y Fd(1)p 1583 4466 V 1583 4507 a(2)1627 4518 y Fn(\).)43 b(Let)32 b(us)g(remark)f(that,)h(for)f Fj(x)d Ff(2)g Fn([0)p Fj(;)3058 4478 y Fi(\024)3099 4455 y Fa(\000)p Fb(\032)p 3058 4494 125 4 v 3102 4552 a Fg(2)3192 4518 y Fn(],)k(for)f(all)f Fj(n)e Ff(2)g Fj(I)-25 b(N)10 b Fn(,)94 4638 y(w)m(e)34 b(ha)m(v)m(e)912 4664 y Fh(\020)962 4760 y Ff(\000)1062 4735 y Fn(~)1039 4760 y Fj(C)1109 4775 y Fg(1)1149 4760 y Fn(\()p Fj(\024)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)1587 4692 y Fd(1)p 1587 4704 31 4 v 1587 4745 a(2)1631 4760 y Fn(\))1669 4664 y Fh(\021)1736 4760 y Fn(exp)1901 4664 y Fh(\020)1951 4760 y Fn(2)p Fj(f)11 b Fn(\(0\)\()p Fj(x)21 b Ff(\000)i Fj(\024)2454 4719 y Fe(\000)p Fi(\032)2549 4760 y Fn(\))2587 4664 y Fh(\021)2665 4760 y Fn(=)k Ff(O)s Fn(\()p Fj(\024)2944 4719 y Fi(n)2991 4760 y Fn(\))p Fj(:)94 4969 y Fn(F)-8 b(rom)32 b(\(3.22\))f(and)i(\(3.23\),)f(for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)1587 4930 y Fi(\024)1628 4907 y Fa(\000)p Fb(\032)p 1587 4946 125 4 v 1631 5004 a Fg(2)1721 4969 y Fn(],)33 b(it)f(results)h(that)508 5263 y Fj(y)556 5278 y Fg(1)595 5263 y Fn(\()p Fj(x)p Fn(\))28 b(=)f Ff(\000)985 5196 y Fj(\024)1041 5159 y Fg(2)1080 5196 y Fj(h)1136 5159 y Fg(3)p 944 5240 272 4 v 944 5331 a Fn(2)p Fj(f)11 b Fn(\(0\))1177 5303 y Fg(2)1243 5263 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))22 b(+)1953 5117 y Fh( )2069 5196 y Fj(\024)2125 5159 y Fg(2)2164 5196 y Fj(h)2220 5159 y Fg(3)p 2029 5240 V 2029 5331 a Fn(2)p Fj(f)11 b Fn(\(0\))2262 5303 y Fg(2)2332 5263 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)2606 5222 y Fg(1)p Fe(\000)p Fg(2)p Fi(\032)2772 5263 y Fn(\))2810 5117 y Fh(!)2893 5263 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)94 5542 y Fn(As)34 b Fj(Y)49 b Ff(\024)28 b Fj(y)498 5557 y Fg(1)537 5542 y Fn(,)k(it)g(results)h(the)g(inequalit)m(y)94 5801 y Fj(Y)22 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\024)487 5734 y Fj(\024)543 5698 y Fg(2)583 5734 y Fj(h)639 5698 y Fg(3)p 447 5778 V 447 5870 a Fn(2)p Fj(f)11 b Fn(\(0\))680 5841 y Fg(2)745 5801 y Fn(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))1303 5705 y Fh(\020)1352 5801 y Fn(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2419 5733 y Fd(1)p 2419 5745 31 4 v 2419 5786 a(2)2460 5760 y Fe(\000)p Fg(2)p Fi(\032)2590 5801 y Fn(\))2628 5705 y Fh(\021)2694 5801 y Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)3211 5734 y(\024)3267 5698 y Fe(\000)p Fi(\032)p 3211 5778 152 4 v 3262 5870 a Fn(2)3372 5801 y(])p Fj(:)199 b Fn(\(3.24\))p eop %%Page: 13 13 13 12 bop 3587 213 a Fn(13)-94 511 y(T)-8 b(aking)32 b Fj(\024)287 526 y Fg(0)359 511 y Fn(smaller,)e(w)m(e)k(get)f(\(3.9\)) f(on)g([0)p Fj(;)17 b(\024)1573 475 y Fe(\000)p Fi(\032)1668 511 y Fn(].)-94 632 y Fk(Step)37 b(3:)50 b(Lo)m(w)m(er)37 b(b)s(ound)i(for)e(Y.)-94 752 y Fn(According)32 b(to)g(\(2.1\))g(and)h (\(2.3\),)f(and)h(as)f Fj(A)1553 716 y Fe(0)1604 752 y Ff(\025)d Fn(0)j(and)h Fj(A)27 b(<)h Fn(0,)k(for)g Fj(x)c Ff(2)h Fj(I)-25 b(R)2713 716 y Fg(+)2771 752 y Fn(,)33 b(w)m(e)h(get)758 964 y(2)p Fj(f)11 b(f)925 928 y Fe(0)948 917 y Fg(2)987 964 y Fj(A)22 b Fn(+)g(2)p Fj(\024)1285 928 y Fg(2)1325 964 y Fj(Af)1457 928 y Fg(2)1496 964 y Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)1849 928 y Fg(3)1888 964 y Fn(\))28 b Ff(\024)g Fn(2)p Fj(\024)2164 928 y Fg(2)2204 964 y Fj(Af)2336 928 y Fg(2)2375 964 y Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)2728 928 y Fg(3)2767 964 y Fn(\))p Fj(:)-94 1172 y Fn(F)-8 b(rom)31 b(\(2.1\))h(and)g(\(2.9\),)g(there)i(exists)f Fj(C)1433 1187 y Fg(3)1505 1172 y Fn(suc)m(h)h(that,)e(for)g Fj(x)c Ff(2)h Fn([0)p Fj(;)17 b(\024)2466 1136 y Fe(\000)p Fi(\032)2560 1172 y Fn(],)33 b(w)m(e)h(ha)m(v)m(e)953 1395 y Ff(\000)p Fn(2)p Fj(\024)1135 1353 y Fg(2)1174 1395 y Fj(Af)1306 1353 y Fg(2)1346 1395 y Fn(\()p Fj(f)f Ff(\000)22 b Fj(f)1623 1353 y Fg(3)1662 1395 y Fn(\))28 b Ff(\024)g Fj(C)1903 1410 y Fg(3)1942 1395 y Fj(\024)2008 1326 y Fd(3)p 2008 1338 31 4 v 2008 1380 a(2)2070 1395 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 1604 y Fn(It)32 b(results)h(that)g(for)f Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1035 1568 y Fe(\000)p Fi(\032)1130 1604 y Fn(],)32 b(w)m(e)i(obtain)d(the)i(inequalit)m(y)742 1827 y(2)p Fj(f)11 b(f)909 1786 y Fe(0)932 1776 y Fg(2)971 1827 y Fj(A)23 b Fn(+)f(2)p Fj(\024)1270 1786 y Fg(2)1309 1827 y Fj(Af)1441 1786 y Fg(2)1480 1827 y Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)1833 1786 y Fg(3)1873 1827 y Fn(\))27 b Ff(\024)h Fj(C)2113 1842 y Fg(3)2153 1827 y Fj(\024)2219 1759 y Fd(3)p 2219 1771 V 2219 1812 a(2)2280 1827 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 2037 y Fn(F)-8 b(rom)31 b(\(2.10\))g(and)i(\(2.27\),)f (it)g(results)h(that)f(there)h(exists)h Fj(C)2150 2052 y Fg(4)2221 2037 y Fn(suc)m(h)g(that)236 2259 y Ff(\000)p Fn(2)p Fj(\024)418 2218 y Fg(2)457 2259 y Fj(Af)11 b Fn(\()p Fj(A)700 2218 y Fg(2)740 2259 y Fj(f)799 2218 y Fg(2)860 2259 y Fn(+)22 b Fj(A)1031 2218 y Fe(0)1054 2207 y Fg(2)1094 2259 y Fn(\))28 b Ff(\025)1265 2163 y Fh(\020)1314 2259 y Fn(4)p Fj(\024)1419 2218 y Fg(2)1459 2259 y Fj(h)1515 2218 y Fg(3)1576 2259 y Fn(+)22 b Fj(C)1744 2274 y Fg(4)1784 2259 y Fj(\024)1840 2218 y Fg(1)p Fe(\000)p Fi(\032)1970 2163 y Fh(\021)2036 2259 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)28 b Fj(x)45 b Ff(2)28 b Fn([0)p Fj(;)17 b(\024)3206 2218 y Fe(\000)p Fi(\032)3301 2259 y Fn(])p Fj(:)-94 2486 y Fn(W)-8 b(e)32 b(w)m(an)m(t)i(compare)e Fj(Y)54 b Fn(with)32 b(the)h(solution)e(in)h Fj(H)1775 2449 y Fg(2)1814 2486 y Fn(\([0)p Fj(;)17 b(\024)2028 2449 y Fe(\000)p Fi(\032)2123 2486 y Fn(]\))32 b(of)-94 2693 y Ff(\000)p Fj(y)31 2708 y Fg(2)70 2652 y Fe(00)135 2693 y Fn(+)22 b(\()p Fj(f)11 b Fn(\(0\))455 2652 y Fg(2)515 2693 y Fn(+)22 b Fj(C)683 2708 y Fg(5)723 2693 y Fj(\024)779 2652 y Fg(1)p Fe(\000)p Fi(\032)909 2693 y Fn(\))p Fj(y)995 2708 y Fg(2)1062 2693 y Fn(=)27 b(\(4)p Fj(\024)1308 2652 y Fg(2)1348 2693 y Fj(h)1404 2652 y Fg(3)1466 2693 y Fn(+)22 b Fj(C)1634 2708 y Fg(4)1673 2693 y Fj(\024)1729 2652 y Fg(1)p Fe(\000)p Fi(\032)1859 2693 y Fn(\))17 b(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b Ff(\000)h Fj(C)2696 2708 y Fg(3)2735 2693 y Fj(\024)2801 2625 y Fd(3)p 2801 2637 V 2801 2678 a(2)2862 2693 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))33 b(\(3.25\))-94 2888 y(satisfying)e(the)i(conditions)-94 3139 y Fj(y)-42 3098 y Fe(0)-46 3164 y Fg(2)-7 3139 y Fn(\(0\))27 b(=)g Fj(Y)327 3098 y Fe(0)350 3139 y Fn(\(0\))p Fj(;)72 b(y)622 3154 y Fg(2)661 3139 y Fn(\()p Fj(\024)755 3098 y Fe(\000)p Fi(\032)850 3139 y Fn(\))28 b(=)1019 2993 y Fh( )1129 3072 y Fj(\024h)p 1095 3116 181 4 v 1095 3216 a Fn(2)1144 3134 y Ff(p)p 1227 3134 49 4 v 82 x Fn(2)1307 3139 y(+)1428 3114 y(~)1405 3139 y Fj(C)7 b(\024)1548 3071 y Fd(3)p 1549 3083 31 4 v 1549 3124 a(4)1593 2993 y Fh(!)1675 3139 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)2179 3098 y Fe(\000)p Fi(\032)2275 3139 y Fn(\))p Fj(:)1096 b Fn(\(3.26\))-94 3400 y(According)28 b(to)h(\(3.12\),)g(\(3.25\))g(and)g(\(3.26\),)g(Lemma)f(2.3)h(with)g Fj(d)e Fn(=)h Fj(\024)2514 3364 y Fe(\000)p Fi(\032)2609 3400 y Fn(,)i Fj(C)k Fn(=)28 b Fj(f)11 b Fn(\(0\))3058 3364 y Fg(2)3112 3400 y Fn(+)k Fj(C)3273 3415 y Fg(5)3312 3400 y Fj(\024)3368 3364 y Fg(1)p Fe(\000)p Fi(\032)3528 3400 y Fn(and)-94 3521 y Fj(u)27 b Fn(=)g Fj(y)140 3536 y Fg(2)201 3521 y Ff(\000)c Fj(Y)e Fn(,)33 b(w)m(e)h(get)e Fj(y)793 3536 y Fg(2)860 3521 y Ff(\024)c Fj(Y)54 b Fn(on)32 b([0)p Fj(;)17 b(\024)1387 3485 y Fe(\000)p Fi(\032)1482 3521 y Fn(].)44 b(W)-8 b(e)33 b(ha)m(v)m(e)-53 3735 y Fj(y)-5 3750 y Fg(2)34 3735 y Fn(\()p Fj(x)p Fn(\))28 b(=)319 3710 y(^)297 3735 y Fj(C)367 3750 y Fg(1)406 3735 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)720 3639 y Fh(\020)770 3735 y Ff(\000)847 3635 y Fh(q)p 930 3635 639 4 v 100 x Fj(f)11 b Fn(\(0\))1114 3706 y Fg(2)1175 3735 y Fn(+)22 b Fj(C)1343 3750 y Fg(5)1382 3735 y Fj(\024)1438 3706 y Fg(1)p Fe(\000)p Fi(\032)1569 3735 y Fj(x)1624 3639 y Fh(\021)1696 3735 y Fn(+)1816 3710 y(^)1794 3735 y Fj(C)1864 3750 y Fg(2)1903 3735 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)2218 3639 y Fh(\020)2267 3635 y(q)p 2350 3635 V 100 x Fj(f)11 b Fn(\(0\))2534 3706 y Fg(2)2595 3735 y Fn(+)22 b Fj(C)2763 3750 y Fg(5)2802 3735 y Fj(\024)2858 3706 y Fg(1)p Fe(\000)p Fi(\032)2989 3735 y Fj(x)3044 3639 y Fh(\021)-53 3961 y Ff(\000)34 3881 y Fn(4)p Fj(\024)139 3845 y Fg(2)179 3881 y Fj(h)235 3845 y Fg(3)291 3881 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g(8)1060 3841 y Fi(C)1110 3850 y Fd(3)p 1060 3857 85 4 v 1060 3915 a Fi(C)1110 3924 y Fd(5)1155 3881 y Fj(\024)1221 3817 y Fd(1)p 1221 3829 31 4 v 1221 3870 a(2)1261 3844 y Fg(+)p Fi(\032)1356 3881 y Fj(f)11 b Fn(\(0\))1540 3845 y Fg(2)1601 3881 y Ff(\000)23 b Fj(C)1771 3896 y Fg(3)1810 3881 y Fj(\024)1876 3817 y Fd(3)p 1876 3829 V 1876 3870 a(2)1943 3881 y Fn(+)f Fj(C)2111 3896 y Fg(4)2150 3881 y Fj(\024)2206 3845 y Fg(1)p Fe(\000)p Fi(\032)2354 3881 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p 34 3938 2910 4 v 1144 4029 a(8)p Fj(f)g Fn(\(0\))1377 4000 y Fg(2)1438 4029 y Ff(\000)23 b Fj(C)1608 4044 y Fg(5)1647 4029 y Fj(\024)1703 4000 y Fg(1)p Fe(\000)p Fi(\032)2970 3961 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)3580 3866 y Fn(\(3.27\))-94 4233 y(F)-8 b(rom)31 b(\(3.5\),)h(w)m(e)h(ha)m(v)m(e)h Fj(\024)846 4196 y Fg(2)886 4233 y Fj(h)942 4196 y Fg(3)1009 4233 y Fn(=)28 b Ff(O)s Fn(\()p Fj(\024)1299 4169 y Fd(1)p 1299 4181 31 4 v 1299 4222 a(2)1343 4233 y Fn(\),)33 b(hence)31 4451 y(4)p Fj(\024)136 4415 y Fg(2)176 4451 y Fj(h)232 4415 y Fg(3)288 4451 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(\024)841 4415 y Fe(\000)p Fi(\032)936 4451 y Fn(\))22 b(+)g(8)1153 4412 y Fi(C)1203 4421 y Fd(3)p 1153 4428 85 4 v 1153 4486 a Fi(C)1203 4495 y Fd(5)1248 4451 y Fj(\024)1314 4387 y Fd(1)p 1314 4399 31 4 v 1314 4440 a(2)1354 4414 y Fg(+)p Fi(\032)1449 4451 y Fj(f)11 b Fn(\(0\))1633 4415 y Fg(2)1694 4451 y Ff(\000)23 b Fj(C)1864 4466 y Fg(3)1903 4451 y Fj(\024)1969 4387 y Fd(3)p 1969 4399 V 1969 4440 a(2)2036 4451 y Fn(+)f Fj(C)2204 4466 y Fg(4)2243 4451 y Fj(\024)2299 4415 y Fg(1)p Fe(\000)p Fi(\032)2446 4451 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(\024)2999 4415 y Fe(\000)p Fi(\032)3094 4451 y Fn(\))p 31 4508 3101 4 v 1199 4600 a Ff(\000)p Fn(8)p Fj(f)g Fn(\(0\))1509 4571 y Fg(2)1570 4600 y Fn(+)22 b Fj(C)1738 4615 y Fg(5)1778 4600 y Fj(\024)1834 4571 y Fg(1)p Fe(\000)p Fi(\032)3170 4531 y Fn(=)27 b Ff(O)s Fn(\()p Fj(\024)3459 4463 y Fd(1)p 3460 4475 31 4 v 3460 4516 a(2)3504 4531 y Fn(\))p Fj(:)3 4798 y Fn(F)-8 b(rom)31 b(\(3.25\))h(and)h(\(3.27\))e(at)i(the)g(p)s(oin)m(t)e Fj(x)d Fn(=)g Fj(\024)1797 4762 y Fe(\000)p Fi(\032)1892 4798 y Fn(,)33 b(it)f(results)g(that)-31 4975 y(^)-53 5000 y Fj(C)17 5015 y Fg(2)56 5000 y Fn(\()p Fj(\024)p Fn(\))c(=)-53 5143 y Ff(\000)47 5118 y Fn(^)24 5143 y Fj(C)94 5158 y Fg(1)134 5143 y Fn(\()p Fj(\024)p Fn(\))17 b(exp\()p Ff(\000)p Fn(2)595 5043 y Fh(q)p 679 5043 639 4 v 679 5143 a Fj(f)11 b Fn(\(0\))863 5114 y Fg(2)924 5143 y Fn(+)22 b Fj(C)1092 5158 y Fg(5)1131 5143 y Fj(\024)1187 5114 y Fg(1)p Fe(\000)p Fi(\032)1317 5143 y Fj(\024)1373 5107 y Fe(\000)p Fi(\032)1469 5143 y Fn(\))g(+)g Ff(O)s Fn(\()p Fj(\024)1813 5079 y Fd(1)p 1813 5091 31 4 v 1813 5132 a(2)1858 5143 y Fn(\))17 b(exp)2078 5047 y Fh(\020)2127 5143 y Ff(\000)2221 5047 y Fh(\020)2271 5043 y(q)p 2354 5043 639 4 v 100 x Fj(f)11 b Fn(\(0\))2538 5114 y Fg(2)2599 5143 y Fn(+)22 b Fj(C)2767 5158 y Fg(5)2806 5143 y Fj(\024)2862 5114 y Fg(1)p Fe(\000)p Fi(\032)3015 5143 y Fn(+)g Fj(f)11 b Fn(\(0\))3297 5047 y Fh(\021)3362 5143 y Fj(\024)3418 5107 y Fe(\000)p Fi(\032)3514 5047 y Fh(\021)3580 5143 y Fj(:)3649 5074 y Fn(\(3.28\))-94 5350 y(F)-8 b(rom)31 b(\(3.27\),)g(w)m(e)j(get)403 5649 y Fj(y)455 5613 y Fe(0)451 5673 y Fg(2)490 5649 y Fn(\(0\))28 b(=)f Ff(\000)833 5569 y Fn(8)892 5529 y Fi(C)942 5538 y Fd(3)p 892 5545 85 4 v 892 5603 a Fi(C)942 5612 y Fd(5)987 5569 y Fj(\024)1053 5505 y Fd(1)p 1053 5517 31 4 v 1053 5558 a(2)1094 5532 y Fg(+)p Fi(\032)1189 5569 y Fj(f)11 b Fn(\(0\))1373 5532 y Fg(3)1434 5569 y Fn(+)22 b(12)p Fj(\024)1686 5532 y Fg(2)1725 5569 y Fj(h)1781 5532 y Fg(3)1821 5569 y Fj(f)11 b Fn(\(0\))21 b(+)h(3)p Fj(\024)2229 5532 y Fg(1)p Fe(\000)p Fi(\032)2360 5569 y Fj(C)2430 5584 y Fg(4)2469 5569 y Fj(f)11 b Fn(\(0\))21 b Ff(\000)i Fj(f)11 b Fn(\(0\))p Fj(C)3028 5584 y Fg(3)3067 5569 y Fj(\024)3133 5505 y Fd(3)p 3133 5517 V 3133 5558 a(2)p 834 5625 2344 4 v 1585 5717 a Fn(\()p Ff(\000)p Fn(8)p Fj(f)g Fn(\(0\))1933 5688 y Fg(2)1994 5717 y Fn(+)22 b Fj(C)2162 5732 y Fg(5)2201 5717 y Fj(\024)2257 5688 y Fg(1)p Fe(\000)p Fi(\032)2388 5717 y Fn(\))403 5849 y Ff(\000)503 5823 y Fn(^)480 5849 y Fj(C)550 5864 y Fg(1)590 5849 y Fn(\()p Fj(\024)p Fn(\))722 5749 y Fh(q)p 805 5749 639 4 v 100 x Fj(f)11 b Fn(\(0\))989 5820 y Fg(2)1050 5849 y Fn(+)22 b Fj(C)1218 5864 y Fg(5)1257 5849 y Fj(\024)1313 5820 y Fg(1)p Fe(\000)p Fi(\032)1466 5849 y Fn(+)1586 5823 y(^)1564 5849 y Fj(C)1634 5864 y Fg(2)1673 5849 y Fn(\()p Fj(\024)p Fn(\))1805 5749 y Fh(q)p 1888 5749 V 100 x Fj(f)11 b Fn(\(0\))2072 5820 y Fg(2)2133 5849 y Fn(+)22 b Fj(C)2301 5864 y Fg(5)2341 5849 y Fj(\024)2397 5820 y Fg(1)p Fe(\000)p Fi(\032)2527 5849 y Fj(:)p eop %%Page: 14 14 14 13 bop 94 213 a Fn(14)94 511 y(The)34 b(condition)d Fj(y)775 475 y Fe(0)771 536 y Fg(2)810 511 y Fn(\(0\))c(=)h Fj(hf)1181 475 y Fe(00)1223 511 y Fn(\(0\),)33 b(\(3.21\))e(and)i (\(3.28\))f(lead)g(to)814 742 y Ff(\000)914 717 y Fn(^)891 742 y Fj(C)961 757 y Fg(1)1001 742 y Fn(\()p Fj(\024)p Fn(\))1133 637 y Fh(q)p 1216 637 639 4 v 105 x Fj(f)11 b Fn(\(0\))1400 713 y Fg(2)1461 742 y Fn(+)22 b Fj(C)1629 757 y Fg(5)1668 742 y Fj(\024)1724 713 y Fg(1)p Fe(\000)p Fi(\032)1877 742 y Fn(+)1985 675 y(3)p Fj(\024)2090 638 y Fg(2)2129 675 y Fj(h)2185 638 y Fg(3)p 1985 719 240 4 v 1989 810 a Fn(2)p Fj(f)11 b Fn(\(0\))2257 742 y(+)22 b Ff(O)s Fn(\()p Fj(\024)2541 674 y Fd(1)p 2541 686 31 4 v 2541 727 a(2)2581 701 y Fg(+)p Fi(\032)2676 742 y Fn(\))28 b(=)g Fj(hf)2961 701 y Fe(00)3003 742 y Fn(\(0\))p Fj(;)94 964 y Fn(hence)1187 1093 y(^)1165 1118 y Fj(C)1235 1133 y Fg(1)1274 1118 y Fn(\()p Fj(\024)p Fn(\))g(=)f Ff(\000)1624 1051 y Fj(hf)1739 1015 y Fe(00)1782 1051 y Fn(\(0\))p 1624 1095 282 4 v 1674 1187 a Fj(f)11 b Fn(\(0\))1939 1118 y(+)2047 1051 y(3)p Fj(\024)2152 1015 y Fg(2)2191 1051 y Fj(h)2247 1015 y Fg(3)p 2047 1095 240 4 v 2056 1187 a Fj(f)g Fn(\(0\))2240 1158 y Fg(2)2319 1118 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)2603 1050 y Fd(1)p 2603 1062 31 4 v 2603 1103 a(2)2643 1077 y Fg(+)p Fi(\032)2739 1118 y Fn(\))p Fj(:)94 1348 y Fn(F)-8 b(ollo)m(wing)30 b(the)j(step)h(1,)e(on)g([0)p Fj(;)1280 1308 y Fi(\024)1321 1285 y Fa(\000)p Fb(\032)p 1280 1324 125 4 v 1325 1382 a Fg(2)1415 1348 y Fn(],)g(w)m(e)i(get)e(the)h(inequalit)m(y)348 1591 y Fj(Y)22 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\025)716 1523 y Fn(3)p Fj(\024)821 1487 y Fg(2)861 1523 y Fj(h)917 1487 y Fg(3)p 701 1567 272 4 v 701 1659 a Fn(2)p Fj(f)11 b Fn(\(0\))934 1630 y Fg(2)999 1591 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))22 b(+)1709 1445 y Fh( )1825 1523 y Fj(\024)1881 1487 y Fg(2)1920 1523 y Fj(h)1976 1487 y Fg(3)p 1785 1567 V 1785 1659 a Fn(2)p Fj(f)11 b Fn(\(0\))2018 1630 y Fg(2)2088 1591 y Ff(\000)2198 1523 y Fj(hf)2313 1487 y Fe(00)2355 1523 y Fn(\(0\))p 2198 1567 282 4 v 2227 1659 a Fj(f)g Fn(\(0\))2411 1630 y Fg(2)2512 1591 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)2796 1522 y Fd(1)p 2796 1534 31 4 v 2796 1576 a(2)2836 1550 y Fg(+)p Fi(\032)2932 1591 y Fn(\))2970 1445 y Fh(!)3052 1591 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)94 1824 y Fn(hence,)35 b(from)c(\(3.21\))94 2045 y Fj(Y)22 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\025)487 1978 y Fj(\024)543 1942 y Fg(2)583 1978 y Fj(h)639 1942 y Fg(3)p 447 2022 272 4 v 447 2114 a Fn(2)p Fj(f)11 b Fn(\(0\))680 2085 y Fg(2)745 2045 y Fn(exp)q(\()p Ff(\000)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))17 b(\(1)22 b Ff(\000)g Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2397 2004 y Fi(\032)2438 2045 y Fn(\)\))16 b Fj(:)1068 b Fn(\(3.29\))94 2263 y(F)-8 b(rom)35 b(\(3.24\))h(and)g(\(3.29\),)h(w)m(e)h(deduce)g(that)e(there)h (exists)g Fj(\024)2421 2278 y Fg(0)2497 2263 y Fn(suc)m(h)h(that,)f (for)f(all)e Fj(\024)h Ff(\024)f Fj(\024)3514 2278 y Fg(0)3554 2263 y Fn(,)j(for)f(all)94 2383 y(pairs)h(\()p Fj(f)5 b(;)17 b(A)p Fn(\))37 b(satisfying)f(\(3.3\),)i(Estimate)e (\(3.9\))h(is)f(true)i(on)f([0)p Fj(;)17 b(\024)2611 2347 y Fe(\000)p Fi(\032)2706 2383 y Fn(].)57 b(The)38 b(pro)s(of)e(of)h(Lemma)e(3.2)94 2504 y(follo)m(ws.)94 2744 y(F)-8 b(rom)32 b(Lemma)f(3.2,)h(w)m(e)i(deduce)g(an)e(estimate)g (for)g Fj(f)2086 2708 y Fe(0)2142 2744 y Fn(in)f(the)i(follo)m(wing)d (prop)s(osition:)94 2906 y Fk(Prop)s(osition)36 b(3.3)49 b Fc(L)-5 b(et)39 b Fj(\032)d Ff(2)g Fn([0)p Fj(;)1382 2867 y Fg(1)p 1382 2883 36 4 v 1382 2940 a(8)1427 2906 y Fn(])p Fc(.)57 b(Ther)-5 b(e)39 b(exists)f Fj(\024)2152 2921 y Fg(0)2231 2906 y Fc(such)h(that,)h(for)f(al)5 b(l)39 b Fj(\024)c Ff(\024)h Fj(\024)3255 2921 y Fg(0)3295 2906 y Fc(,)k(for)f(al)5 b(l)39 b(p)-5 b(airs)94 3026 y Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))35 b Fc(solutions)g(of)f Fn(\()p Fj(GL)p Fn(\))1117 3041 y Fe(1)1227 3026 y Fc(satisfying)h (\(3.3\),)f(we)g(have)g(the)h(estimate)94 3201 y Fj(f)153 3159 y Fe(0)177 3201 y Fn(\()p Fj(x)p Fn(\))27 b(=)h Fj(\024)495 3159 y Fg(2)535 3201 y Fj(h)591 3159 y Fg(2)647 3104 y Fh(\020)696 3201 y Fn(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)1131 3114 y(p)p 1214 3114 49 4 v 87 x Fn(2)p Fj(x)p Fn(\))g(+)f Ff(O)s Fn(\()p Fj(\024)1653 3159 y Fi(\032)1693 3201 y Fn(\))1731 3104 y Fh(\021)1797 3201 y Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)17 b(\024)2360 3159 y Fe(\000)p Fi(\032)2455 3201 y Fn(])p Fj(:)1116 b Fn(\(3.30\))94 3362 y Fk(Pro)s(of)94 3483 y Fn(According)33 b(to)f(Prop)s(osition)f(2.5)h(and)h(Lemma)e(3.2,)h(w) m(e)i(get)644 3708 y Fj(f)703 3667 y Fe(0)726 3708 y Fn(\()p Fj(x)p Fn(\))28 b(=)1038 3640 y Fj(\024)1094 3604 y Fg(2)1134 3640 y Fj(h)1190 3604 y Fg(3)p 998 3684 272 4 v 998 3776 a Fn(2)p Fj(f)11 b Fn(\(0\))1231 3747 y Fg(2)1296 3708 y Fn(\(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2391 3667 y Fi(\032)2431 3708 y Fn(\)\))17 b Fj(:)2568 3587 y Fh(\022)2642 3640 y Fn(1)p 2639 3684 57 4 v 2639 3776 a Fj(h)2727 3708 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)3011 3639 y Fd(3)p 3011 3651 31 4 v 3011 3693 a(2)3052 3667 y Fe(\000)p Fg(2)p Fi(\032)3182 3708 y Fn(\))3220 3587 y Fh(\023)3298 3708 y Fj(:)94 3956 y Fn(F)-8 b(or)32 b Fj(f)11 b Fn(\(0\))27 b(=)623 3917 y Fg(1)p 594 3933 95 4 v 594 3942 a Fe(p)p 653 3942 36 3 v 56 x Fg(2)720 3956 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)1004 3892 y Fd(1)p 1004 3904 31 4 v 1004 3946 a(4)1049 3956 y Fn(\),)32 b(and)h(for)f Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1838 3920 y Fe(\000)p Fi(\032)1933 3956 y Fn(],)33 b(w)m(e)g(ha)m(v)m(e)1125 4165 y(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))28 b(=)g(exp)q(\()p Ff(\000)2111 4079 y(p)p 2194 4079 49 4 v 86 x Fn(2)p Fj(x)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2642 4097 y Fd(1)p 2642 4109 31 4 v 2642 4150 a(4)2683 4124 y Fe(\000)p Fi(\032)2778 4165 y Fn(\))p Fj(:)94 4337 y Fn(As)34 b Fj(\032)28 b Ff(2)g Fn([0)p Fj(;)540 4298 y Fg(1)p 540 4314 36 4 v 540 4371 a(8)585 4337 y Fn(],)33 b(Estimate)f(\(3.30\))g(follo)m(ws)f(and)i(this)f(ac)m (hiev)m(ed)i(the)f(pro)s(of)f(of)g(Prop)s(osition)f(3.3.)94 4499 y Fk(Remark)37 b(3.4)49 b Fc(L)-5 b(et)42 b(us)g(r)-5 b(e)g(c)g(al)5 b(l)41 b(that)g(in)g([12])g(\(se)-5 b(e)41 b(also)g([15]\),)h(we)f(have)g(obtaine)-5 b(d)40 b(an)h(asymptotic)94 4619 y(matche)-5 b(d)40 b(solution)h(of)f Fn(\()p Fj(GL)p Fn(\))1195 4634 y Fe(1)1311 4619 y Fc(denote)-5 b(d)40 b(by)h Fn(\()p Fj(f)1903 4583 y Fi(v)r(d;)p Fg(\()p Fi(n)p Fg(\))2097 4619 y Fj(;)17 b(A)2214 4583 y Fi(v)r(d;)p Fg(\()p Fi(n)p Fg(\))2409 4619 y Fn(\))p Fc(.)62 b(F)-7 b(or)40 b Fj(n)f Fn(=)f(1)p Fc(,)k(at)f(the)g(sup)-5 b(erhe)g(ating)94 4739 y(\014eld,)35 b(we)f(have)g(shown)g(that)94 4919 y Fj(f)153 4877 y Fi(v)r(d;)p Fg(\(1\))340 4919 y Fn(\()p Fj(x)p Fn(\))28 b(=)g(tanh\()856 4851 y Fj(\024x)p 846 4895 132 4 v 846 4913 a Ff(p)p 929 4913 49 4 v 82 x Fn(2)1010 4919 y(+)22 b Fj(x)1163 4934 y Fg(0)1202 4919 y Fn(\))h(+)1371 4851 y Fj(\024)p 1371 4895 57 4 v 1375 4987 a Fn(4)1453 4919 y(exp)q(\()p Ff(\000)1717 4832 y(p)p 1801 4832 49 4 v 1801 4919 a Fn(2)p Fj(x)p Fn(\))p Fj(;)1655 b Fn(\(3.31\))94 5141 y Fc(wher)-5 b(e)35 b Fn(tanh\()p Fj(x)658 5156 y Fg(0)698 5141 y Fn(\))27 b(=)906 5101 y Fg(1)p 877 5117 95 4 v 877 5127 a Fe(p)p 936 5127 36 3 v 55 x Fg(2)981 5141 y Fc(.)45 b(It)34 b(r)-5 b(esults)35 b(that,)h(for)e Fj(x)28 b Ff(2)g Fn([0)p Fj(;)17 b(\024)2203 5104 y Fe(\000)p Fi(\032)2298 5141 y Fn(])p Fc(,)35 b(we)g(have)94 5397 y Fn(\()p Fj(f)191 5356 y Fi(v)r(d;)p Fg(\(1\))378 5397 y Fn(\))416 5356 y Fe(0)439 5397 y Fn(\()p Fj(x)p Fn(\))28 b(=)712 5248 y Ff(p)p 795 5248 49 4 v 82 x Fn(2)p 712 5374 132 4 v 753 5465 a(4)854 5397 y Fj(\024)927 5301 y Fh(\020)976 5397 y Fn(1)22 b Ff(\000)g Fn(exp)r(\()p Ff(\000)1411 5310 y(p)p 1494 5310 49 4 v 87 x Fn(2)p Fj(x)p Fn(\))1636 5301 y Fh(\021)1708 5397 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)1982 5356 y Fg(2)p Fe(\000)p Fi(\032)2112 5397 y Fn(\))p Fj(:)1448 b Fn(\(3.32\))94 5647 y Fc(A)-5 b(c)g(c)g(or)g(ding)38 b(to)g(\(1.9\))e(and)i(\(1.10\))e(with)i Fj(f)11 b Fn(\(0\))32 b(=)1992 5579 y(1)p 1950 5623 132 4 v 1950 5641 a Ff(p)p 2033 5641 49 4 v 82 x Fn(2)2092 5647 y Fc(,)38 b(the)g(main)f(term)h (of)f(Estimate)h(\(3.30\))f(is)g(the)94 5820 y(one)e(of)f(Estimate)h (\(3.32\).)44 b(Ther)-5 b(efor)g(e,)34 b(we)g(susp)-5 b(e)g(ct)35 b(that)g(Estimate)g(\(3.30\))f(is)g(optimal.)p eop %%Page: 15 15 15 14 bop 3587 213 a Fn(15)-94 511 y Fk(3.3.)56 b(Estimate)35 b(for)j Fj(A)855 475 y Fe(0)900 511 y Fn(+)22 b Fj(Af)3 632 y Fn(T)-8 b(o)36 b(get)f(an)g(estimate)g(for)g Fj(A)1072 596 y Fe(0)1120 632 y Fn(+)24 b Fj(Af)11 b Fn(,)36 b(w)m(e)g(use)h (Lemma)d(2.3)h(and)g(estimates)g(for)g Fj(A)3093 596 y Fe(0)3117 632 y Fn(,)h Fj(A)p Fn(,)g Fj(f)3375 596 y Fe(0)3434 632 y Fn(and)f Fj(f)-94 752 y Fn(obtained)d(in)f(Prop)s (ositions)h(2.4)g(and)h(2.5.)-94 947 y Fk(Lemma)k(3.5)49 b Fc(L)-5 b(et)36 b Fj(\032)29 b Ff(2)h Fn([0)p Fj(;)968 907 y Fg(1)p 968 923 36 4 v 968 981 a(8)1014 947 y Fn(])p Fc(.)47 b(Ther)-5 b(e)35 b(exists)h Fj(\024)1723 962 y Fg(0)1798 947 y Fc(such)f(that,)i(for)e(al)5 b(l)35 b Fj(\024)30 b Ff(\024)g Fj(\024)2796 962 y Fg(0)2835 947 y Fc(,)36 b(for)f(al)5 b(l)36 b(p)-5 b(airs)35 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))-94 1067 y Fc(solutions)34 b(of)h Fn(\()p Fj(GL)p Fn(\))648 1082 y Fe(1)757 1067 y Fc(satisfying)g(\(3.3\),)f(we)g(have)g(the)h(fol)5 b(lowing)34 b(estimate)-94 1306 y Fj(A)-21 1265 y Fe(0)2 1306 y Fn(+)p Fj(Af)k Fn(=)351 1238 y(1)p 351 1283 49 4 v 351 1374 a(2)426 1306 y(exp)q(\()p Ff(\000)742 1238 y Fn(1)p 700 1283 132 4 v 700 1300 a Ff(p)p 784 1300 49 4 v 784 1382 a Fn(2)842 1306 y Fj(x)p Fn(\))952 1185 y Fh(\022)1013 1306 y Ff(\000)p Fj(\024)1146 1265 y Fg(2)1187 1306 y Fj(h)1243 1265 y Fg(3)1299 1306 y Fn(exp)q(\()p Ff(\000)1563 1219 y(p)p 1646 1219 V 87 x Fn(2)p Fj(x)p Fn(\))22 b Ff(\000)1948 1238 y Fn(1)p 1920 1283 105 4 v 1920 1374 a(8)p Fj(h)2057 1306 y Fn(+)g(3)p Fj(\024)2260 1265 y Fg(2)2299 1306 y Fj(h)2355 1265 y Fg(3)2417 1306 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2701 1237 y Fd(1)p 2701 1249 31 4 v 2701 1291 a(2)2742 1265 y Fg(+)p Fi(\032)2837 1306 y Fn(\))2875 1185 y Fh(\023)2953 1306 y Fj(;)99 b Ff(8)28 b Fj(x)g Ff(2)h Fn([0)p Fj(;)17 b(\024)3516 1265 y Fe(\000)p Fi(\032)3610 1306 y Fn(])p Fj(:)p Fn(\(3.33\))3 1548 y Fk(Pro)s(of)37 b Fn(.)-94 1668 y Fk(Step)g(1:)50 b(Upp)s(er)38 b(b)s(ound)h(for)e Fj(A)1239 1632 y Fe(0)1285 1668 y Fn(+)22 b Fj(Af)-94 1788 y Fn(W)-8 b(e)38 b(in)m(tro)s(duce)g (the)g(function)g Fj(Z)44 b Fn(:=)37 b Fj(A)1403 1752 y Fe(0)1452 1788 y Fn(+)26 b Fj(Af)11 b Fn(.)60 b(F)-8 b(rom)36 b(\(2.5\),)j(this)f(function)f(is)h(p)s(ositiv)m(e.)59 b(A)m(t)39 b(the)-94 1909 y(p)s(oin)m(t)34 b Fj(x)g Fn(=)f Fj(\024)417 1873 y Fe(\000)p Fi(\032)512 1909 y Fn(,)k(from)e(\(2.9\),) h(\(2.10\))f(and)h(\(2.27\),)f(w)m(e)i(deduce)h(that)d(there)i(exists) 3102 1884 y(^)3079 1909 y Fj(C)43 b Fn(and)3407 1884 y(~)3385 1909 y Fj(C)g Fn(suc)m(h)-94 2029 y(that)-72 2198 y(^)-94 2223 y Fj(C)6 b(\024)48 2155 y Fd(1)p 49 2167 V 49 2208 a(2)89 2182 y Fe(\000)p Fg(2)p Fi(\032)236 2223 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(\024)740 2182 y Fe(\000)p Fi(\032)835 2223 y Fn(\))28 b Ff(\024)g Fj(Z)7 b Fn(\()p Fj(\024)1174 2182 y Fe(\000)p Fi(\032)1269 2223 y Fn(\))28 b(=)f(\()p Fj(A)1549 2182 y Fe(0)1595 2223 y Fn(+)22 b Fj(Af)11 b Fn(\)\()p Fj(\024)1957 2182 y Fe(\000)p Fi(\032)2052 2223 y Fn(\))27 b Ff(\024)2245 2198 y Fn(~)2223 2223 y Fj(C)7 b(\024)2366 2155 y Fd(1)p 2366 2167 V 2366 2208 a(2)2406 2182 y Fe(\000)p Fg(2)p Fi(\032)2553 2223 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)3057 2182 y Fe(\000)p Fi(\032)3152 2223 y Fn(\))p Fj(:)219 b Fn(\(3.34\))-94 2418 y(F)-8 b(rom)31 b(\(1)p Fj(:)p Fn(2\))363 2433 y Fi(b)396 2418 y Fn(,)i(w)m(e)h(ha)m(v)m(e)1307 2538 y Fj(Z)1381 2497 y Fe(0)1431 2538 y Fn(=)28 b Fj(Af)1667 2497 y Fg(2)1728 2538 y Fn(+)22 b Fj(A)1899 2497 y Fe(0)1923 2538 y Fj(f)33 b Fn(+)22 b Fj(Af)2234 2497 y Fe(0)2257 2538 y Fj(:)-94 2708 y Fn(Di\013eren)m(tiating)29 b(more,)j(and)h (using)f(the)h(\()p Fj(G:L:)p Fn(\))g(equations,)g(w)m(e)h(get)545 2918 y Fj(Z)619 2876 y Fe(00)689 2918 y Fn(=)27 b Fj(A)865 2876 y Fe(0)889 2918 y Fj(f)948 2876 y Fg(2)1009 2918 y Fn(+)22 b(2)p Fj(Af)1288 2876 y Fe(0)1311 2918 y Fj(f)33 b Fn(+)22 b Fj(Af)1622 2876 y Fg(3)1683 2918 y Fn(+)g(2)p Fj(A)1903 2876 y Fe(0)1926 2918 y Fj(f)1985 2876 y Fe(0)2030 2918 y Fn(+)g Fj(\024)2184 2876 y Fg(2)2224 2918 y Fj(A)p Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)2650 2876 y Fg(3)2712 2918 y Fn(+)g Fj(A)2883 2876 y Fg(2)2922 2918 y Fj(f)11 b Fn(\))p Fj(:)-94 3127 y Fn(It)32 b(results)h(that)g (the)g(function)f Fj(Z)39 b Fn(satis\014es)33 b(the)g(equation)-94 3321 y Ff(\000)p Fj(Z)57 3280 y Fe(00)121 3321 y Fn(+)23 b(\()p Fj(f)317 3280 y Fg(2)378 3321 y Fn(+)f(2)p Fj(f)584 3280 y Fe(0)607 3321 y Fn(\))p Fj(Z)34 b Fn(=)28 b Ff(\000)p Fj(\024)983 3280 y Fg(2)1023 3321 y Fj(A)p Fn(\()p Ff(\000)p Fj(f)33 b Fn(+)22 b Fj(f)1449 3280 y Fg(3)1510 3321 y Fn(+)g Fj(A)1681 3280 y Fg(2)1721 3321 y Fj(f)11 b Fn(\))p Fj(:)1591 b Fn(\(3.35\))-94 3516 y(F)-8 b(rom)31 b(\(1)p Fj(:)p Fn(2\))363 3531 y Fi(b)397 3516 y Fn(,)i(w)m(e)h(ha)m(v)m(e)h Fj(Z)901 3480 y Fe(0)924 3516 y Fn(\(0\))28 b(=)h Fj(f)11 b Fn(\(0\)\()p Fj(h)22 b Fn(+)g Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\)\))28 b Ff(\025)h Fn(0)p Fj(:)k Fn(F)-8 b(rom)31 b(\(2.4\))i(at)g(the)g(p)s(oin)m(t)f Fj(x)d Fn(=)g(0,)k(w)m(e)-94 3636 y(get)1086 3810 y Fj(h)22 b Fn(+)g Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))27 b(=)1872 3743 y(\(1)22 b Ff(\000)g Fj(f)11 b Fn(\(0\))2264 3707 y Fg(2)2303 3743 y Fn(\))2341 3707 y Fg(2)p 1784 3787 684 4 v 1784 3878 a Fn(2\()p Fj(h)22 b Ff(\000)h Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\)\))2478 3810 y Fj(:)-94 4036 y Fn(F)-8 b(rom)31 b(\(2.5\))h(and)g(\(2.10\))g (at)g(the)h(p)s(oin)m(t)f Fj(x)c Fn(=)g(0,)k(w)m(e)i(get)969 4259 y Fj(h)p Fn(\(2)22 b(+)g Ff(O)s Fn(\()p Fj(\024)1418 4190 y Fd(3)p 1418 4202 31 4 v 1418 4243 a(2)1459 4217 y Fe(\000)p Fg(2)p Fi(\032)1589 4259 y Fn(\)\))28 b Ff(\024)g Fj(h)22 b Ff(\000)h Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\))26 b Ff(\024)j Fn(2)p Fj(h:)-94 4468 y Fn(It)j(results)h(that)327 4644 y Fj(f)11 b Fn(\(0\)\(1)22 b Ff(\000)g Fj(f)11 b Fn(\(0\))903 4608 y Fg(2)942 4644 y Fn(\))980 4608 y Fg(2)p 327 4688 693 4 v 621 4780 a Fn(4)p Fj(h)1057 4712 y Ff(\024)28 b Fj(Z)1236 4671 y Fe(0)1259 4712 y Fn(\(0\))g(=)f Fj(f)11 b Fn(\(0\))17 b(\()o Fj(h)23 b Fn(+)f Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\)\))26 b Ff(\024)2518 4644 y Fj(f)11 b Fn(\(0\)\(1)21 b Ff(\000)i Fj(f)11 b Fn(\(0\))3094 4608 y Fg(2)3132 4644 y Fn(\))3170 4608 y Fg(2)p 2491 4688 745 4 v 2491 4794 a Fn(4)p Fj(h)p Fn(\(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)2989 4731 y Fd(3)p 2989 4743 31 4 v 2989 4784 a(2)3030 4758 y Fe(\000)p Fg(2)p Fi(\032)3160 4794 y Fn(\)\))3246 4712 y Fj(:)-94 5048 y Fn(Hence,)34 b(as)e Fj(f)11 b Fn(\(0\))27 b(=)708 4981 y(1)p 667 5025 132 4 v 667 5043 a Ff(p)p 750 5043 49 4 v 82 x Fn(2)831 5048 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1115 4980 y Fd(1)p 1115 4992 31 4 v 1115 5033 a(4)1159 5048 y Fn(\),)33 b(w)m(e)h(get)-94 5353 y Fj(Z)-20 5311 y Fe(0)3 5353 y Fn(\(0\))27 b(=)h Fj(f)11 b Fn(\(0\))17 b(\()o Fj(h)22 b Fn(+)g Fj(A)p Fn(\(0\))p Fj(f)11 b Fn(\(0\)\))27 b(=)1351 5285 y(1)p 1233 5329 286 4 v 1233 5429 a(16)1331 5347 y Ff(p)p 1414 5347 49 4 v 82 x Fn(2)o Fj(h)1545 5256 y Fh(\020)1595 5353 y Fn(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)1950 5284 y Fd(1)p 1950 5296 31 4 v 1950 5337 a(4)1994 5353 y Fn(\))2032 5256 y Fh(\021)2098 5353 y Fj(:)1311 b Fn(\(3.36\))-94 5604 y(No)m(w,)33 b(w)m(e)i(can)e(estimate)f(the)i(term)e Ff(\000)p Fj(\024)1405 5568 y Fg(2)1445 5604 y Fj(A)p Fn(\()p Ff(\000)p Fj(f)i Fn(+)22 b Fj(f)1872 5568 y Fg(3)1934 5604 y Fn(+)g Fj(A)2105 5568 y Fg(2)2145 5604 y Fj(f)11 b Fn(\).)45 b(First,)32 b(for)h Fj(x)c Ff(2)g Fj(I)-25 b(R)3001 5568 y Fg(+)3059 5604 y Fn(,)34 b(as)f Fj(A)c(<)f Fn(0)33 b(and)-94 5725 y(0)27 b Ff(\024)h Fj(f)39 b Ff(\024)28 b Fn(1,)k(w)m(e)i(ha)m(v)m(e)1353 5845 y Ff(\000)p Fj(\024)1486 5804 y Fg(2)1527 5845 y Fj(A)p Fn(\()p Ff(\000)p Fj(f)f Fn(+)22 b Fj(f)1953 5804 y Fg(3)1992 5845 y Fn(\))28 b Fj(<)f Fn(0)p Fj(:)p eop %%Page: 16 16 16 15 bop 94 213 a Fn(16)94 511 y(According)37 b(to)g(\(2.6\),)g (\(2.9\))f(and)h(using)g Fj(f)11 b Fn(\(0\))34 b(=)2010 472 y Fg(1)p 1981 488 95 4 v 1981 498 a Fe(p)p 2040 498 36 3 v 55 x Fg(2)2110 511 y Fn(+)25 b Ff(O)s Fn(\()p Fj(\024)2397 448 y Fd(1)p 2397 460 31 4 v 2397 501 a(4)2442 511 y Fn(\),)38 b(w)m(e)g(deduce)g(that)f(there)g(exists)h Fj(C)3835 526 y Fg(6)94 646 y Fn(suc)m(h)d(that,)d(for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1055 609 y Fe(\000)p Fi(\032)1150 646 y Fn(],)33 b(w)m(e)g(get)g(the)g(estimate)1089 860 y Ff(\000)p Fj(\024)1222 819 y Fg(2)1262 860 y Fj(A)1335 819 y Fg(3)1375 860 y Fj(f)38 b Ff(\024)1566 764 y Fh(\020)1616 860 y Fn(2)p Fj(\024)1721 819 y Fg(2)1760 860 y Fj(h)1816 819 y Fg(3)1878 860 y Fn(+)22 b Fj(C)2046 875 y Fg(6)2085 860 y Fj(\024)2151 792 y Fd(3)p 2151 804 V 2151 845 a(4)2196 764 y Fh(\021)2262 860 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)94 1078 y Fn(T)-8 b(o)40 b(get)f(an)g(upp)s(er)h(b)s(ound)f(for)g Fj(Z)7 b Fn(,)41 b(as)e Fj(f)1632 1041 y Fg(2)1698 1078 y Fn(+)26 b(2)p Fj(f)1908 1041 y Fe(0)1970 1078 y Ff(\025)39 b Fj(f)11 b Fn(\(0\))2270 1041 y Fg(2)2309 1078 y Fn(,)41 b(w)m(e)f(compare)f(this)g(function)g(with)g(the)94 1198 y(solution)32 b(in)f Fj(H)666 1162 y Fg(2)705 1198 y Fn(\([0)p Fj(;)17 b(\024)919 1162 y Fe(\000)p Fi(\032)1014 1198 y Fn(]\))33 b(of)f(the)h(equation)94 1398 y Ff(\000)p Fj(V)251 1357 y Fe(00)315 1398 y Fn(+)22 b Fj(f)11 b Fn(\(0\))597 1357 y Fg(2)636 1398 y Fj(V)49 b Fn(=)846 1302 y Fh(\020)895 1398 y Fn(2)p Fj(\024)1000 1357 y Fg(2)1040 1398 y Fj(h)1096 1357 y Fg(3)1157 1398 y Fn(+)22 b Fj(C)1325 1413 y Fg(6)1365 1398 y Fj(\024)1431 1330 y Fd(3)p 1431 1342 V 1431 1383 a(4)1475 1302 y Fh(\021)1541 1398 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)1467 b Fn(\(3.37\))94 1585 y(and)33 b(satisfying)94 1773 y Fj(V)173 1731 y Fe(0)196 1773 y Fn(\(0\))28 b(=)f Fj(Z)526 1731 y Fe(0)549 1773 y Fn(\(0\))p Fj(;)72 b(V)22 b Fn(\()p Fj(\024)946 1731 y Fe(\000)p Fi(\032)1041 1773 y Fn(\))28 b(=)1232 1747 y(~)1210 1773 y Fj(C)7 b(\024)1353 1704 y Fd(1)p 1353 1716 V 1353 1758 a(2)1394 1731 y Fe(\000)p Fg(2)p Fi(\032)1541 1773 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)2045 1731 y Fe(\000)p Fi(\032)2140 1773 y Fn(\))p Fj(:)1420 b Fn(\(3.38\))94 1960 y(F)-8 b(rom)32 b(\(3.34\),)f(\(3.35\),)h (\(3.37\),)g(\(3.38\))g(and)h(as)g Fj(Z)h Ff(\025)28 b Fn(0,)33 b(w)m(e)g(get)g(on)f([0)p Fj(;)17 b(\024)2801 1924 y Fe(\000)p Fi(\032)2896 1960 y Fn(])1298 2161 y Ff(\000)p Fn(\()p Fj(Z)30 b Ff(\000)22 b Fj(V)g Fn(\))1726 2120 y Fe(00)1791 2161 y Fn(+)g Fj(f)11 b Fn(\(0\))2073 2120 y Fg(2)2111 2161 y Fn(\()p Fj(Z)29 b Ff(\000)23 b Fj(V)f Fn(\))27 b Ff(\024)h Fn(0)p Fj(;)94 2362 y Fn(and)1189 2482 y(\()p Fj(Z)h Ff(\000)22 b Fj(V)g Fn(\))1539 2441 y Fe(0)1562 2482 y Fn(\(0\))28 b(=)f(0)p Fj(;)100 b Fn(\()p Fj(Z)29 b Ff(\000)22 b Fj(V)g Fn(\)\()p Fj(\024)2438 2441 y Fe(\000)p Fi(\032)2533 2482 y Fn(\))28 b Ff(\024)g Fn(0)p Fj(:)94 2648 y Fn(W)-8 b(e)33 b(apply)g(Lemma)e(2.3)h(with)g Fj(d)c Fn(=)f Fj(\024)1496 2612 y Fe(\000)p Fi(\032)1591 2648 y Fn(,)33 b Fj(C)i Fn(=)27 b Fj(f)11 b Fn(\(0\))2043 2612 y Fg(2)2115 2648 y Fn(and)32 b Fj(u)27 b Fn(=)h Fj(Z)h Ff(\000)23 b Fj(V)54 b Fn(and)32 b(get)h Fj(Z)i Ff(\024)28 b Fj(V)21 b Fn(.)94 2769 y(F)-8 b(or)32 b Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)622 2733 y Fe(\000)p Fi(\032)717 2769 y Fn(],)33 b(the)g(solution)e(of)h(\(3.37\))g(is)g (giv)m(en)h(b)m(y)136 2953 y Fj(V)22 b Fn(\()p Fj(x)p Fn(\))136 3139 y(=)262 3113 y(~)240 3139 y Fj(C)310 3154 y Fg(3)349 3139 y Fn(\()p Fj(\024)p Fn(\))17 b(exp\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))23 b(+)1181 3113 y(~)1159 3139 y Fj(C)1229 3154 y Fg(4)1268 3139 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)q(\()p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))21 b Ff(\000)2002 2993 y Fh( )2118 3071 y Fj(\024)2174 3035 y Fg(2)2214 3071 y Fj(h)2270 3035 y Fg(3)p 2078 3115 272 4 v 2078 3207 a Fn(4)p Fj(f)11 b Fn(\(0\))2311 3178 y Fg(2)2381 3139 y Fn(+)2571 3071 y Fj(C)2641 3086 y Fg(6)p 2489 3115 V 2489 3207 a Fn(8)p Fj(f)g Fn(\(0\))2722 3178 y Fg(2)2771 3139 y Fj(\024)2837 3070 y Fd(3)p 2837 3082 31 4 v 2837 3124 a(4)2882 2993 y Fh(!)2964 3139 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)g Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)3625 3078 y Fn(\(3.39\))94 3388 y(F)-8 b(rom)32 b(\(3.38\))f(and)i(\(3.39\),)f(w)m(e)i(get)117 3563 y(~)94 3588 y Fj(C)164 3603 y Fg(4)204 3588 y Fn(\()p Fj(\024)p Fn(\))27 b(=)467 3492 y Fh(\020)517 3588 y Ff(\000)616 3563 y Fn(~)594 3588 y Fj(C)664 3603 y Fg(3)703 3588 y Fn(\()p Fj(\024)p Fn(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)1141 3520 y Fd(1)p 1142 3532 V 1142 3573 a(2)1182 3547 y Fe(\000)p Fg(2)p Fi(\032)1312 3588 y Fn(\))1350 3492 y Fh(\021)1417 3588 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(\024)1970 3547 y Fe(\000)p Fi(\032)2064 3588 y Fn(\))p Fj(:)1496 b Fn(\(3.40\))94 3788 y(F)-8 b(rom)32 b(\(3.39\),)f(w)m(e)j(deduce)g(that)745 4037 y Fj(V)824 3996 y Fe(0)847 4037 y Fn(\(0\))27 b(=)h Ff(\000)p Fj(f)11 b Fn(\(0\))1386 4012 y(~)1364 4037 y Fj(C)1434 4052 y Fg(3)1473 4037 y Fn(\()p Fj(\024)p Fn(\))22 b(+)g Fj(f)11 b Fn(\(0\))1931 4012 y(~)1909 4037 y Fj(C)1979 4052 y Fg(4)2018 4037 y Fn(\()p Fj(\024)p Fn(\))22 b(+)2372 3970 y(3)p 2280 4014 233 4 v 2280 4106 a(4)p Fj(f)11 b Fn(\(0\))2522 4037 y Fj(\024)2578 3996 y Fg(2)2618 4037 y Fj(h)2674 3996 y Fg(3)2735 4037 y Fn(+)2880 3970 y(3)p Fj(C)2999 3985 y Fg(6)p 2843 4014 V 2843 4106 a Fn(8)p Fj(f)g Fn(\(0\))3085 4037 y Fj(\024)3151 3969 y Fd(3)p 3152 3981 31 4 v 3152 4022 a(4)3196 4037 y Fj(:)94 4294 y Fn(F)-8 b(rom)38 b(\(3.40\))g(and)h(taking)g(accoun)m(t)g(the)h (condition)d Fj(V)2203 4258 y Fe(0)2226 4294 y Fn(\(0\))i(=)f Fj(Z)2578 4258 y Fe(0)2601 4294 y Fn(\(0\))h(\(see)h(\(3.36\)\))e(and)h (replacing)94 4424 y Fj(f)11 b Fn(\(0\))32 b(with)572 4384 y Fg(1)p 543 4400 95 4 v 543 4410 a Fe(p)p 601 4410 36 3 v 601 4465 a Fg(2)669 4424 y Fn(+)22 b Ff(O)s Fn(\()p Fj(\024)953 4360 y Fd(1)p 953 4372 31 4 v 953 4413 a(4)997 4424 y Fn(\),)33 b(w)m(e)h(get)117 4669 y(~)94 4694 y Fj(C)164 4709 y Fg(3)204 4694 y Fn(\()p Fj(\024)p Fn(\))27 b(=)h Ff(\000)607 4627 y Fn(1)p 554 4671 154 4 v 554 4763 a(16)p Fj(h)735 4598 y Fh(\020)784 4694 y Fn(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)1139 4626 y Fd(1)p 1139 4638 31 4 v 1139 4679 a(4)1184 4694 y Fn(\))1222 4598 y Fh(\021)1294 4694 y Fn(+)1402 4627 y(3)p Fj(\024)1507 4591 y Fg(2)1546 4627 y Fj(h)1602 4591 y Fg(3)p 1402 4671 240 4 v 1497 4763 a Fn(2)1651 4694 y Fj(:)1947 b Fn(\(3.41\))94 4938 y(W)-8 b(e)38 b(ha)m(v)m(e)g(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))35 b(=)g(exp)q(\()p Ff(\000)1486 4898 y Fg(1)p 1457 4914 95 4 v 1457 4924 a Fe(p)p 1516 4924 36 3 v 55 x Fg(2)1561 4938 y Fj(x)p Fn(\))1671 4841 y Fh(\020)1721 4938 y Fn(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)2076 4874 y Fd(1)p 2076 4886 31 4 v 2076 4927 a(4)2116 4901 y Fe(\000)p Fi(\032)2211 4938 y Fn(\))2249 4841 y Fh(\021)2336 4938 y Fn(on)36 b([0)p Fj(;)17 b(\024)2651 4901 y Fe(\000)p Fi(\032)2746 4938 y Fn(].)56 b(F)-8 b(rom)36 b(\(3.39\),)h(\(3.40\))f(and)94 5090 y(\(3.41\),)c(w)m(e)i (get)f(the)g(estimate,)e(for)h Fj(x)d Ff(2)f Fn([0)p Fj(;)1751 5051 y Fi(\024)1792 5027 y Fa(\000)p Fb(\032)p 1751 5067 125 4 v 1796 5124 a Fg(2)1886 5090 y Fn(])136 5331 y Fj(V)22 b Fn(\()p Fj(x)p Fn(\))28 b(=)487 5264 y(1)p 487 5308 49 4 v 487 5399 a(2)562 5331 y(exp)q(\()p Ff(\000)878 5264 y Fn(1)p 836 5308 132 4 v 836 5326 a Ff(p)p 920 5326 49 4 v 920 5408 a Fn(2)979 5331 y Fj(x)p Fn(\))1089 5210 y Fh(\022)1150 5331 y Ff(\000)p Fj(\024)1283 5290 y Fg(2)1323 5331 y Fj(h)1379 5290 y Fg(3)1435 5331 y Fn(exp)q(\()p Ff(\000)1699 5244 y(p)p 1782 5244 V 87 x Fn(2)p Fj(x)p Fn(\))23 b Ff(\000)2084 5264 y Fn(1)p 2056 5308 105 4 v 2056 5399 a(8)p Fj(h)2193 5331 y Fn(+)f(3)p Fj(\024)2396 5290 y Fg(2)2436 5331 y Fj(h)2492 5290 y Fg(3)2553 5331 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2837 5263 y Fd(3)p 2838 5275 31 4 v 2838 5316 a(4)2878 5290 y Fe(\000)p Fi(\032)2973 5331 y Fn(\))3011 5210 y Fh(\023)3089 5331 y Fj(:)3625 5335 y Fn(\(3.42\))94 5573 y(As)34 b Fj(Z)g Ff(\024)28 b Fj(V)54 b Fn(on)33 b([0)p Fj(;)17 b(\024)868 5537 y Fe(\000)p Fi(\032)963 5573 y Fn(],)33 b(w)m(e)g(deduce)h(that)f (there)g(exists)g Fj(\024)2304 5588 y Fg(0)2376 5573 y Fn(suc)m(h)h(that,)f(for)f(all)e Fj(\024)e Ff(\024)g Fj(\024)3364 5588 y Fg(0)3404 5573 y Fn(,)33 b(w)m(e)g(get)136 5805 y Fj(Z)7 b Fn(\()p Fj(x)p Fn(\))28 b Ff(\024)484 5737 y Fn(1)p 484 5781 49 4 v 484 5873 a(2)559 5805 y(exp)q(\()p Ff(\000)875 5737 y Fn(1)p 833 5781 132 4 v 833 5799 a Ff(p)p 917 5799 49 4 v 917 5881 a Fn(2)976 5805 y Fj(x)p Fn(\))1086 5683 y Fh(\022)1147 5805 y Ff(\000)p Fj(\024)1280 5763 y Fg(2)1320 5805 y Fj(h)1376 5763 y Fg(3)1432 5805 y Fn(exp)q(\()p Ff(\000)1696 5717 y(p)p 1779 5717 V 87 x Fn(2)1828 5805 y Fj(x)p Fn(\))23 b Ff(\000)2081 5737 y Fn(1)p 2053 5781 105 4 v 2053 5873 a(8)p Fj(h)2190 5805 y Fn(+)f(3)p Fj(\024)2393 5763 y Fg(2)2433 5805 y Fj(h)2489 5763 y Fg(3)2550 5805 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2834 5736 y Fd(3)p 2835 5748 31 4 v 2835 5789 a(4)2875 5763 y Fe(\000)p Fi(\032)2970 5805 y Fn(\))3008 5683 y Fh(\023)3086 5805 y Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)17 b(\024)3649 5763 y Fe(\000)p Fi(\032)3744 5805 y Fn(])p Fj(:)3839 5809 y Fn(\(3.43\))p eop %%Page: 17 17 17 16 bop 3587 213 a Fn(17)-94 511 y Fk(Step)37 b(2:)50 b(Lo)m(w)m(er)37 b(b)s(ound)i(for)e Fj(A)1223 475 y Fe(0)1269 511 y Fn(+)22 b Fj(Af)-94 632 y Fn(First,)31 b(from)h(\(2.10\),)f(w)m (e)j(remark)e(that)h(there)g(exists)g Fj(C)1986 647 y Fg(7)2058 632 y Fn(suc)m(h)h(that,)f(for)f Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)3019 596 y Fe(\000)p Fi(\032)3114 632 y Fn(],)988 820 y Ff(\000)p Fj(\024)1121 779 y Fg(2)1161 820 y Fj(A)p Fn(\()p Ff(\000)p Fj(f)34 b Fn(+)22 b Fj(f)1588 779 y Fg(3)1627 820 y Fn(\))27 b Ff(\025)i Fj(C)1868 835 y Fg(7)1907 820 y Fj(\024)1973 751 y Fd(3)p 1973 763 31 4 v 1973 805 a(2)2034 820 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)-94 994 y Fn(According)27 b(to)g(\(2.3\))f(and)i(\(2.6\),)g(w)m(e)g(deduce)h(that)e(there)i (exists)f Fj(C)2389 1009 y Fg(8)2455 994 y Fn(suc)m(h)h(that)e Fj(f)2935 958 y Fg(2)2986 994 y Fn(+)12 b(2)p Fj(f)3182 958 y Fe(0)3232 994 y Ff(\024)28 b Fn(\()p Fj(f)11 b Fn(\(0\))3559 958 y Fg(2)3609 994 y Fn(+)-94 1115 y Fj(C)-24 1130 y Fg(8)15 1115 y Fj(\024)71 1079 y Fg(1)p Fe(\000)p Fi(\032)201 1115 y Fn(\))p Fj(:)33 b Fn(T)-8 b(o)33 b(get)f(a)g(lo)m(w) m(er)h(b)s(ound,)g(w)m(e)h(compare)e Fj(Z)39 b Fn(with)32 b(the)h(solution)f(of)-94 1192 y Fh(8)-94 1267 y(<)-94 1416 y(:)21 1300 y Ff(\000)p Fj(W)204 1264 y Fe(00)269 1300 y Fn(+)22 b(\()p Fj(f)11 b Fn(\(0\))589 1264 y Fg(2)649 1300 y Fn(+)22 b Fj(C)817 1315 y Fg(8)857 1300 y Fj(\024)913 1264 y Fg(1)p Fe(\000)p Fi(\032)1043 1300 y Fn(\))p Fj(W)42 b Fn(=)1318 1204 y Fh(\020)1368 1300 y Fn(2)p Fj(\024)1473 1264 y Fg(2)1512 1300 y Fj(h)1568 1264 y Fg(3)1630 1300 y Fn(+)22 b Fj(C)1798 1315 y Fg(6)1837 1300 y Fj(\024)1903 1236 y Fd(3)p 1903 1248 V 1903 1290 a(4)1948 1204 y Fh(\021)2014 1300 y Fn(exp)q(\()p Ff(\000)p Fn(3)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)g Fj(C)2794 1315 y Fg(7)2833 1300 y Fj(\024)2899 1236 y Fd(3)p 2899 1248 V 2899 1290 a(2)2960 1300 y Fn(exp)r(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(;)21 1446 y(W)127 1410 y Fe(0)150 1446 y Fn(\(0\))27 b(=)h Fj(Z)480 1410 y Fe(0)503 1446 y Fn(\(0\))p Fj(;)99 b(W)14 b Fn(\()p Fj(\024)954 1410 y Fe(\000)p Fi(\032)1049 1446 y Fn(\))28 b(=)1240 1421 y(^)1218 1446 y Fj(C)7 b(\024)1361 1382 y Fd(1)p 1361 1394 V 1361 1435 a(2)1402 1409 y Fe(\000)p Fg(2)p Fi(\032)1549 1446 y Fn(exp)q(\()p Ff(\000)p Fj(f)k Fn(\(0\))p Fj(\024)2053 1410 y Fe(\000)p Fi(\032)2148 1446 y Fn(\))p Fj(:)3580 1366 y Fn(\(3.44\))-94 1622 y(F)-8 b(ollo)m(wing)40 b(the)k(step)g(1,)i(w)m(e)f(apply)e(Lemma) f(\(2.3\))h(with)g Fj(d)j Fn(=)g Fj(\024)2417 1586 y Fe(\000)p Fi(\032)2512 1622 y Fn(,)h Fj(C)53 b Fn(=)46 b Fj(f)11 b Fn(\(0\))3015 1586 y Fg(2)3083 1622 y Fn(+)30 b Fj(C)3259 1637 y Fg(8)3298 1622 y Fj(\024)3354 1586 y Fg(1)p Fe(\000)p Fi(\032)3528 1622 y Fn(and)-94 1743 y Fj(u)d Fn(=)g Fj(W)36 b Ff(\000)23 b Fj(Z)7 b Fn(,)32 b(and)h(w)m(e)g(get)g Fj(W)41 b Ff(\024)29 b Fj(Z)39 b Fn(on)32 b([0)p Fj(;)17 b(\024)1605 1707 y Fe(\000)p Fi(\032)1700 1743 y Fn(].)44 b(The)33 b(function)g Fj(W)46 b Fn(is)32 b(equal)g(to)-53 1927 y Fj(W)14 b Fn(\()p Fj(x)p Fn(\))28 b(=)337 1902 y(^)315 1927 y Fj(C)385 1942 y Fg(3)424 1927 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)q(\()p Ff(\000)837 1827 y Fh(q)p 921 1827 639 4 v 921 1927 a Fj(f)11 b Fn(\(0\))1105 1898 y Fg(2)1165 1927 y Fn(+)22 b Fj(C)1333 1942 y Fg(8)1373 1927 y Fj(\024)1429 1898 y Fg(1)p Fe(\000)p Fi(\032)1559 1927 y Fj(x)p Fn(\))h(+)1795 1902 y(^)1773 1927 y Fj(C)1843 1942 y Fg(4)1882 1927 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)q(\()2218 1827 y Fh(q)p 2301 1827 V 100 x Fj(f)11 b Fn(\(0\))2485 1898 y Fg(2)2546 1927 y Fn(+)22 b Fj(C)2714 1942 y Fg(8)2753 1927 y Fj(\024)2809 1898 y Fg(1)p Fe(\000)p Fi(\032)2939 1927 y Fj(x)p Fn(\))-53 2149 y Ff(\000)34 2069 y Fn(2)p Fj(\024)139 2033 y Fg(2)179 2069 y Fj(h)235 2033 y Fg(3)291 2069 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b Ff(\000)h Fn(8)1062 2030 y Fi(C)1112 2039 y Fd(7)p 1062 2046 85 4 v 1062 2103 a Fi(C)1112 2112 y Fd(8)1156 2069 y Fj(\024)1222 2005 y Fd(1)p 1222 2017 31 4 v 1222 2058 a(2)1263 2032 y Fg(+)p Fi(\032)1358 2069 y Fj(f)11 b Fn(\(0\))1542 2033 y Fg(2)1603 2069 y Fn(+)22 b Fj(C)1771 2084 y Fg(7)1810 2069 y Fj(\024)1876 2005 y Fd(3)p 1876 2017 V 1876 2058 a(2)1943 2069 y Fn(+)g Fj(C)2111 2084 y Fg(6)2150 2069 y Fj(\024)2206 2033 y Fg(1)p Fe(\000)p Fi(\032)2354 2069 y Fn(exp)q(\()p Ff(\000)p Fn(2)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p 34 2126 2910 4 v 1144 2217 a(8)p Fj(f)g Fn(\(0\))1377 2189 y Fg(2)1438 2217 y Ff(\000)23 b Fj(C)1608 2232 y Fg(8)1647 2217 y Fj(\024)1703 2189 y Fg(1)p Fe(\000)p Fi(\032)2970 2149 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)3580 2056 y Fn(\(3.45\))-94 2371 y(F)-8 b(rom)31 b(\(3.45\))g(at)i(the)g(p)s (oin)m(t)e Fj(x)d Fn(=)g Fj(\024)1228 2335 y Fe(\000)p Fi(\032)1356 2371 y Fn(and)k(from)g(\(3.44\),)g(it)f(results)i(that)-31 2517 y(^)-53 2543 y Fj(C)17 2558 y Fg(4)56 2543 y Fn(\()p Fj(\024)p Fn(\))-53 2685 y(=)28 b Ff(\000)150 2660 y Fn(^)128 2685 y Fj(C)198 2700 y Fg(3)237 2685 y Fn(\()p Fj(\024)p Fn(\))17 b(exp)q(\()p Ff(\000)p Fn(2)699 2585 y Fh(q)p 782 2585 600 4 v 100 x Fj(f)11 b Fn(\(0\))22 b(+)g Fj(C)1156 2700 y Fg(8)1195 2685 y Fj(\024)1251 2657 y Fg(1)p Fe(\000)p Fi(\032)1382 2685 y Fj(\024)1438 2649 y Fe(\000)p Fi(\032)1533 2685 y Fn(\))g(+)g Ff(O)s Fn(\()p Fj(\024)1877 2621 y Fd(1)p 1877 2633 31 4 v 1877 2675 a(2)1922 2685 y Fn(\))17 b(exp)2142 2589 y Fh(\020)2191 2685 y Ff(\000)p Fn(\()2306 2585 y Fh(q)p 2390 2585 600 4 v 2390 2685 a Fj(f)11 b Fn(\(0\))21 b(+)h Fj(C)2763 2700 y Fg(8)2803 2685 y Fj(\024)2859 2657 y Fg(1)p Fe(\000)p Fi(\032)3011 2685 y Fn(+)g Fj(f)11 b Fn(\(0\)\))p Fj(\024)3387 2649 y Fe(\000)p Fi(\032)3482 2589 y Fh(\021)3548 2685 y Fj(:)3617 2617 y Fn(\(3.46\))-94 2919 y(As)33 b Fj(W)156 2883 y Fe(0)179 2919 y Fn(\(0\))27 b(=)g Fj(Z)508 2883 y Fe(0)532 2919 y Fn(\(0\))32 b(and)g Fj(f)11 b Fn(\(0\))28 b(=)1245 2852 y(1)p 1203 2896 132 4 v 1203 2914 a Ff(p)p 1286 2914 49 4 v 82 x Fn(2)1367 2919 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1651 2851 y Fd(1)p 1651 2863 31 4 v 1651 2904 a(4)1696 2919 y Fn(\),)32 b(w)m(e)i(get)e(from)g(\(3.46\))g(and)g (follo)m(wing)e(the)j(step)h(1)-72 3183 y(^)-94 3208 y Fj(C)-24 3223 y Fg(3)15 3208 y Fn(\()p Fj(\024)p Fn(\))27 b(=)h Ff(\000)418 3141 y Fn(1)p 365 3185 154 4 v 365 3277 a(16)p Fj(h)546 3112 y Fh(\020)595 3208 y Fn(1)22 b(+)g Ff(O)s Fn(\()p Fj(\024)950 3140 y Fd(1)p 950 3152 31 4 v 950 3193 a(4)995 3208 y Fn(\))1033 3112 y Fh(\021)1105 3208 y Fn(+)1213 3141 y(3)p Fj(\024)1318 3105 y Fg(2)1357 3141 y Fj(h)1413 3105 y Fg(3)p 1213 3185 240 4 v 1308 3277 a Fn(2)1485 3208 y(+)g Ff(O)s Fn(\()p Fj(\024)1769 3140 y Fd(1)p 1769 3152 31 4 v 1769 3193 a(2)1809 3167 y Fg(+)p Fi(\032)1904 3208 y Fn(\))p Fj(:)1467 b Fn(\(3.47\))-94 3424 y(On)32 b([0)p Fj(;)198 3384 y Fi(\024)239 3361 y Fa(\000)p Fb(\032)p 198 3400 125 4 v 242 3458 a Fg(2)332 3424 y Fn(],)h(according)f(to)g(\(3.45\),)g(\(3.46\))g(and)h(\(3.47\),) e(w)m(e)j(get)f(that)f Fj(W)46 b Fn(is)32 b(giv)m(en)h(b)m(y)486 3585 y(1)p 486 3629 49 4 v 486 3720 a(2)561 3652 y(exp)q(\()p Ff(\000)877 3585 y Fn(1)p 835 3629 132 4 v 835 3647 a Ff(p)p 919 3647 49 4 v 919 3729 a Fn(2)977 3652 y Fj(x)p Fn(\))1087 3531 y Fh(\022)1148 3652 y Ff(\000)p Fj(\024)1281 3611 y Fg(2)1322 3652 y Fj(h)1378 3611 y Fg(3)1434 3652 y Fn(exp)q(\()p Ff(\000)1698 3565 y(p)p 1781 3565 V 87 x Fn(2)p Fj(x)p Fn(\))22 b Ff(\000)2083 3585 y Fn(1)p 2055 3629 105 4 v 2055 3720 a(8)p Fj(h)2192 3652 y Fn(+)g(3)p Fj(\024)2395 3611 y Fg(2)2434 3652 y Fj(h)2490 3611 y Fg(3)2552 3652 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2836 3584 y Fd(1)p 2836 3596 31 4 v 2836 3637 a(2)2877 3611 y Fg(+)p Fi(\032)2972 3652 y Fn(\))3010 3531 y Fh(\023)3088 3652 y Fj(:)-94 3900 y Fn(Hence,)34 b(as)e Fj(W)42 b Ff(\024)28 b Fj(Z)39 b Fn(on)33 b([0)p Fj(;)17 b(\024)999 3863 y Fe(\000)p Fi(\032)1094 3900 y Fn(],)32 b(w)m(e)i(get)f(the)g(estimate)e (for)h Fj(x)c Ff(2)g Fn([0)p Fj(;)2501 3860 y Fi(\024)2542 3837 y Fa(\000)p Fb(\032)p 2501 3876 125 4 v 2546 3934 a Fg(2)2636 3900 y Fn(])-85 4050 y(1)p -85 4095 49 4 v -85 4186 a(2)-9 4118 y(exp)q(\()p Ff(\000)307 4050 y Fn(1)p 265 4095 132 4 v 265 4113 a Ff(p)p 348 4113 49 4 v 82 x Fn(2)407 4118 y Fj(x)p Fn(\))517 3997 y Fh(\022)578 4118 y Ff(\000)p Fj(\024)711 4077 y Fg(2)751 4118 y Fj(h)807 4077 y Fg(3)863 4118 y Fn(exp)q(\()p Ff(\000)1127 4031 y(p)p 1211 4031 V 1211 4118 a Fn(2)p Fj(x)p Fn(\))22 b Ff(\000)1513 4050 y Fn(1)p 1485 4095 105 4 v 1485 4186 a(8)p Fj(h)1622 4118 y Fn(+)g(3)p Fj(\024)1825 4077 y Fg(2)1864 4118 y Fj(h)1920 4077 y Fg(3)1982 4118 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2266 4050 y Fd(1)p 2266 4062 31 4 v 2266 4103 a(2)2306 4077 y Fg(+)p Fi(\032)2402 4118 y Fn(\))2440 3997 y Fh(\023)2528 4118 y Ff(\024)28 b Fj(Z)7 b Fn(\()p Fj(x)p Fn(\))p Fj(:)571 b Fn(\(3.48\))-94 4335 y(T)-8 b(aking)27 b(accoun)m(t)i(\(3.43\))e(and)h(\(3.48\),)g(w)m (e)h(deduce)h(that)d(there)i(exists)g Fj(\024)2581 4350 y Fg(0)2648 4335 y Fn(suc)m(h)g(that,)g(for)f(all)d Fj(\024)j Ff(\024)g Fj(\024)3618 4350 y Fg(0)3658 4335 y Fn(,)-94 4456 y(for)c(all)f(pairs)i(\()p Fj(f)5 b(;)17 b(A)p Fn(\))25 b(satisfying)f(\(3.3\),)i(w)m(e)g(get)g(Estimate)e(\(3.33\))g(on)h([0)p Fj(;)17 b(\024)2635 4420 y Fe(\000)p Fi(\032)2730 4456 y Fn(].)41 b(The)26 b(pro)s(of)f(of)f(Lemma)-94 4576 y(3.5)32 b(follo)m(ws.)-94 4817 y(Then,)46 b(from)41 b(Lemma)h(3.5,)j(w)m(e)e(can)g(impro)m(v)m(e)g(Inequalit)m(y)f(\(2.5\)) g(and)h(get)g(the)g(follo)m(wing)d(prop)s(o-)-94 4937 y(sition:)-94 5102 y Fk(Prop)s(osition)35 b(3.6)49 b Fc(Ther)-5 b(e)35 b(exists)h Fj(\024)1308 5117 y Fg(0)1384 5102 y Fc(such)g(that,)g(for)g(al)5 b(l)36 b Fj(\024)30 b Ff(\024)g Fj(\024)2384 5117 y Fg(0)2424 5102 y Fc(,)36 b(for)g(al)5 b(l)36 b(p)-5 b(airs)35 b Fn(\()p Fj(f)5 b(;)17 b(A)p Fn(\))36 b Fc(solutions)-94 5222 y(of)e Fn(\()p Fj(GL)p Fn(\))239 5237 y Fe(1)349 5222 y Fc(satisfying)g (\(3.3\),)g(we)h(have)f(the)h(fol)5 b(lowing)33 b(estimate)-94 5387 y Ff(\000)p Fj(Af)39 b Fn(=)27 b Fj(A)319 5346 y Fe(0)365 5387 y Ff(\000)22 b Fj(B)5 b(;)72 b Ff(8)28 b Fj(x)h Ff(2)f Fn([0)p Fj(;)17 b(\024)1079 5346 y Fe(\000)p Fi(\032)1174 5387 y Fn(])p Fj(;)2208 b Fn(\(3.49\))-94 5551 y Fc(wher)-5 b(e)34 b Fj(B)40 b Fc(is)34 b(de\014ne)-5 b(d)34 b(by)-94 5755 y Fj(B)5 b Fn(\()p Fj(x)p Fn(\))28 b(:=)284 5687 y(1)p 284 5731 49 4 v 284 5823 a(2)360 5755 y(exp)q(\()p 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Fn(2)p Fj(x)p Fn(\))p Fj(;)100 b Ff(8)28 b Fj(x)g Ff(2)g Fn([0)p Fj(;)17 b(\024)3362 2614 y Fe(\000)p Fi(\032)3457 2655 y Fn(])p Fj(:)114 b Fn(\(3.53\))94 2949 y Fk(Pro)s(of)94 3069 y Fn(F)-8 b(rom)32 b(Inequalities)f(\(2.9\))h(and)h(\(2.26\),)f (for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)2105 3033 y Fe(\000)p Fi(\032)2200 3069 y Fn(],)33 b(w)m(e)g(get)293 3359 y Fj(A)366 3323 y Fe(0)411 3359 y Ff(\000)23 b Fj(Af)38 b Ff(\024)775 3263 y Fh(\020)825 3359 y Fj(h)22 b Fn(+)g Ff(O)s Fn(\()p Fj(\024)1187 3295 y Fd(1)p 1188 3307 31 4 v 1188 3348 a(2)1228 3322 y Fe(\000)p Fg(2)p Fi(\032)1358 3359 y Fn(\))1396 3263 y Fh(\021)1462 3359 y Fn(exp)r(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))22 b(+)2197 3291 y Fj(h)p 2134 3336 184 4 v 2134 3427 a(f)11 b Fn(\(0\))2344 3263 y Fh(\020)2393 3359 y Fj(f)g Fn(\(0\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2873 3318 y Fg(1)p Fe(\000)p Fi(\032)3003 3359 y Fn(\))3041 3263 y Fh(\021)3108 3359 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p 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Fj(x)p Fn(\))28 b Ff(\024)g Fj(A)3115 4360 y Fe(0)3161 4401 y Ff(\000)22 b Fj(Af)5 b(:)212 b Fn(\(3.55\))94 4689 y(F)-8 b(rom)32 b(\(3.54\))f(and)i(\(3.55\),)f(for)g Fj(x)c Ff(2)g Fn([0)p Fj(;)17 b(\024)1633 4653 y Fe(\000)p Fi(\032)1728 4689 y Fn(])p Fj(;)33 b Fn(w)m(e)g(deduce)i(the)e (estimate)94 4932 y Fj(A)167 4891 y Fe(0)213 4932 y Ff(\000)23 b Fj(Af)38 b Fn(=)28 b(2)642 4836 y Fh(\020)691 4932 y Fj(h)22 b Fn(+)g Ff(O)s Fn(\()p Fj(\024)1053 4864 y Fd(1)p 1053 4876 31 4 v 1053 4917 a(2)1094 4891 y Fe(\000)p Fg(2)p Fi(\032)1224 4932 y Fn(\))1262 4836 y Fh(\021)1328 4932 y Fn(exp)q(\()p Ff(\000)p Fj(f)11 b Fn(\(0\))p Fj(x)p Fn(\))p Fj(:)1729 b Fn(\(3.56\))94 5175 y(F)-8 b(rom)32 b(\(3.56\))f(and)i(Lemma)e(3.5)h(\(see)i(\(3.33\)\),)e(w)m(e)h(can)g (write)685 5412 y Fj(S)751 5376 y Fg(2)818 5412 y Fn(=)27 b(\()p Fj(A)1032 5376 y Fe(0)1078 5412 y Ff(\000)22 b Fj(Af)11 b Fn(\))p Fj(:)p Fn(\()p Fj(A)1485 5376 y Fe(0)1531 5412 y Fn(+)22 b Fj(Af)11 b Fn(\))685 5579 y(=)798 5512 y(1)p 798 5556 49 4 v 798 5648 a(8)874 5483 y Fh(\020)923 5579 y Ff(\000)p Fn(8)p Fj(\024)1105 5538 y Fg(2)1145 5579 y Fj(h)1201 5538 y Fg(4)1257 5579 y Fn(exp)q(\()p Ff(\000)1521 5493 y(p)p 1605 5493 V 1605 5579 a Fn(2)o Fj(x)p Fn(\))23 b Ff(\000)g Fn(1)f(+)g(24)p Fj(\024)2192 5538 y Fg(2)2231 5579 y Fj(h)2287 5538 y Fg(4)2348 5579 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2622 5538 y Fi(\032)2663 5579 y Fn(\))2701 5483 y Fh(\021)2767 5579 y Fn(exp)q(\()p Ff(\000)3031 5493 y(p)p 3115 5493 V 3115 5579 a Fn(2)o Fj(x)p Fn(\))p Fj(:)94 5845 y Fn(The)34 b(pro)s(of)e(of)g(Prop)s (osition)f(3.8)h(follo)m(ws.)p eop %%Page: 19 19 19 18 bop 3587 213 a Fn(19)-94 511 y Fk(4.)56 b(Pro)s(of)37 b(of)h(Theorem)f(1.7)3 709 y Fn(In)h(this)f(section,)i(\014rst,)g (using)f(Prop)s(ositions)e(3.3,)i(3.6)f(and)h(3.8,)g(w)m(e)h(get)e (Inequalit)m(y)h(1.7)f(for)g(all)-94 830 y(pairs)d(\()p Fj(f)5 b(;)17 b(A)p Fn(\))34 b(solution)f(of)h(\()p Fj(GL)p Fn(\))1130 845 y Fe(1)1240 830 y Fn(suc)m(h)i(that)e Fj(f)11 b Fn(\(0\))31 b Ff(2)h Fn([)2054 791 y Fg(1)p 2025 806 95 4 v 2025 816 a Fe(p)p 2084 816 36 3 v 55 x Fg(2)2152 830 y Ff(\000)24 b Fj(\013)q(\024)2382 766 y Fd(1)p 2382 778 31 4 v 2382 819 a(4)2427 830 y Fj(;)2510 791 y Fg(1)p 2480 806 95 4 v 2480 816 a Fe(p)p 2539 816 36 3 v 55 x Fg(2)2608 830 y Fn(+)f Fj(\013)q(\024)2836 766 y Fd(1)p 2836 778 31 4 v 2836 819 a(4)2881 830 y Fn(].)50 b(In)35 b(the)g(follo)m(wing,)-94 950 y(w)m(e)e(use)h(the)f (elemen)m(tary)f(equalit)m(y)-8 b(,)32 b(for)g Fj(a)c(>)g Fn(0)k(and)h Fj(b)28 b(>)g Fn(0)-94 1103 y Ff(p)p -11 1103 292 4 v 87 x Fj(a)40 1161 y Fg(2)101 1190 y Fn(+)22 b Fj(b)240 1161 y Fg(2)308 1190 y Fn(=)27 b Fj(a)c Fn(+)f Fj(b)g Ff(\000)1039 1122 y Fn(2)p Fj(ab)p 756 1167 708 4 v 756 1266 a(a)g Fn(+)g Fj(b)h Fn(+)1089 1185 y Ff(p)p 1172 1185 292 4 v 81 x Fj(a)1223 1238 y Fg(2)1285 1266 y Fn(+)f Fj(b)1424 1238 y Fg(2)1474 1190 y Fj(:)1984 b Fn(\(4.1\))-94 1439 y Fk(Step)37 b(1:)50 b(New)37 b(estimate)f(for)i Fj(A)1253 1402 y Fe(0)1276 1439 y Fn(\(0\))-94 1559 y(According)32 b(to)g(\(3.6\))g(and)h(Prop)s(osition)e(3.6)h(\(see)h(\(3.49\)\),)f(w)m (e)i(get)e(the)h(upp)s(er)h(b)s(ound)e(for)g Fj(A)3411 1523 y Fe(0)3435 1559 y Fn(\(0\))3560 1523 y Fg(2)762 1829 y Fj(h)818 1788 y Fg(2)885 1829 y Ff(\024)c Fn(2)1056 1712 y Fh(Z)1139 1738 y Fe(1)1102 1901 y Fg(0)1230 1829 y Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(A)1473 1788 y Fe(0)1496 1829 y Fn(\()p Fj(t)p Fn(\))1607 1788 y Fg(2)1647 1829 y Fj(dt)22 b Ff(\000)g Fn(2)1920 1712 y Fh(Z)2003 1738 y Fi(\024)2044 1715 y Fa(\000)p Fb(\032)1966 1901 y Fg(0)2148 1829 y Fj(A)2221 1788 y Fe(0)2245 1829 y Fn(\()p Fj(t)p Fn(\))p Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(B)5 b Fn(\()p Fj(t)p Fn(\))p Fj(dt;)-94 2074 y Fn(where)33 b Fj(B)38 b Fn(is)32 b(de\014ned)i(in)e(\(3.50\).)43 b(F)-8 b(rom)31 b(\(3.8\),)h(w)m(e)h(get)-94 2329 y Fj(h)-38 2288 y Fg(2)29 2329 y Ff(\024)28 b(\000)p Fn(2)277 2212 y Fh(Z)360 2238 y Fi(\024)401 2215 y Fa(\000)p Fb(\032)323 2401 y Fg(0)505 2329 y Fj(A)578 2288 y Fe(0)602 2329 y Fn(\()p Fj(t)p Fn(\))p Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(B)5 b Fn(\()p Fj(t)p Fn(\))p Fj(dt)22 b Fn(+)1279 2212 y Fh(Z)1362 2238 y Fg(+)p Fe(1)1325 2401 y Fg(0)1508 2329 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b Ff(\000)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))2097 2288 y Fg(4)2136 2329 y Fn(\))p Fj(dx)22 b Ff(\000)h Fn(6)p Fj(\024)2507 2288 y Fe(\000)p Fg(2)2618 2212 y Fh(Z)2701 2238 y Fg(+)p Fe(1)2664 2401 y Fg(0)2847 2329 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)3096 2288 y Fe(0)3119 2329 y Fn(\()p Fj(x)p Fn(\))3250 2288 y Fg(2)3290 2329 y Fj(dx:)62 b Fn(\(4.2\))-94 2563 y(According)32 b(to)g(\(2.4\))g(and)h(\(4.1\))f(with)g(the)h(c)m(hoice)g Fj(a)28 b Fn(=)f Fj(\024)2062 2527 y Fe(\000)p Fg(1)2157 2563 y Fj(f)2216 2527 y Fe(0)2271 2563 y Fn(and)33 b Fj(b)28 b Fn(=)g Fj(S)6 b Fn(,)32 b(it)g(results)h(the)g(equalit)m(y) -43 2736 y(1)p -85 2780 132 4 v -85 2798 a Ff(p)p -2 2798 49 4 v 82 x Fn(2)57 2804 y(\(1)22 b Ff(\000)h Fj(f)325 2763 y Fg(2)364 2804 y Fn(\))k(=)h Fj(\024)589 2763 y Fe(\000)p Fg(1)684 2804 y Fj(f)743 2763 y Fe(0)788 2804 y Fn(+)22 b Fj(S)28 b Ff(\000)22 b Fj(T)8 b(;)2320 b Fn(\(4.3\))-94 3054 y(where)33 b Fj(T)47 b Fn(is)32 b(de\014ned)i(on)e Fj(I)-25 b(R)953 3018 y Fg(+)1044 3054 y Fn(b)m(y)-94 3304 y Fj(T)14 b Fn(\()p Fj(x)p Fn(\))27 b(:=)700 3236 y(2)p Fj(\024)805 3200 y Fe(\000)p Fg(1)899 3236 y Fj(f)958 3200 y Fe(0)981 3236 y Fj(S)p 276 3280 1195 4 v 276 3397 a(S)h Fn(+)22 b Fj(\024)518 3368 y Fe(\000)p Fg(1)612 3397 y Fj(f)671 3368 y Fe(0)716 3397 y Fn(+)814 3297 y Fh(q)p 897 3297 574 4 v 100 x Fj(S)963 3368 y Fg(2)1025 3397 y Fn(+)g Fj(\024)1179 3368 y Fe(\000)p Fg(2)1273 3397 y Fn(\()p Fj(f)1370 3368 y Fe(0)1393 3397 y Fn(\))1431 3368 y Fg(2)1481 3304 y Fj(:)1977 b Fn(\(4.4\))-94 3602 y(F)-8 b(rom)31 b(\(4.3\))h(and)g(follo)m(wing)e([5])j(\(see)g(Prop)s (osition)e(3.1\),)h(w)m(e)i(get)377 3740 y Fh(Z)460 3766 y Fe(1)423 3929 y Fg(0)534 3857 y Fn(\(1)22 b Ff(\000)h Fj(f)802 3816 y Fg(4)841 3857 y Fn(\()p Fj(x)p Fn(\)\))p Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Fn(=)1513 3790 y(1)p 1448 3834 181 4 v 1448 3934 a(2)1497 3852 y Ff(p)p 1579 3852 49 4 v 1579 3934 a Fn(2)1638 3857 y Fj(\024)1694 3816 y Fe(\000)p Fg(1)1789 3857 y Fn(\(1)21 b Ff(\000)i Fj(f)11 b Fn(\(0\))2181 3816 y Fg(2)2220 3857 y Fn(\)\(3)22 b(+)g Fj(f)11 b Fn(\(0\))2649 3816 y Fg(2)2687 3857 y Fn(\))377 4067 y(+)453 3985 y Ff(p)p 536 3985 V 82 x Fn(2)601 3950 y Fh(Z)684 3976 y Fe(1)647 4138 y Fg(0)775 4067 y Fj(f)g Fn(\()p Fj(x)p Fn(\)\(1)22 b(+)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1362 4026 y Fg(2)1402 4067 y Fn(\))p Fj(:S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)22 b Ff(\000)1892 3980 y(p)p 1975 3980 V 87 x Fn(2)2040 3950 y Fh(Z)2123 3976 y Fe(1)2086 4138 y Fg(0)2214 4067 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b(+)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))2801 4026 y Fg(2)2841 4067 y Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx;)-94 4307 y Fn(and)476 4317 y Fh(Z)559 4344 y Fe(1)522 4506 y Fg(0)650 4434 y Fj(f)d Fn(\()p Fj(x)p Fn(\))p Fj(f)899 4393 y Fe(0)922 4434 y Fn(\()p Fj(x)p Fn(\))1053 4393 y Fg(2)1093 4434 y Fj(dx)28 b Fn(=)1402 4367 y Fj(\024)p 1340 4411 181 4 v 1340 4511 a Fn(4)1389 4429 y Ff(p)p 1472 4429 49 4 v 82 x Fn(2)1531 4434 y(\(1)22 b Ff(\000)g Fj(f)11 b Fn(\(0\))1923 4393 y Fg(2)1962 4434 y Fn(\))2000 4393 y Fg(2)2062 4434 y Ff(\000)22 b Fj(\024)2234 4317 y Fh(Z)2317 4344 y Fe(1)2280 4506 y Fg(0)2408 4434 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)2657 4393 y Fe(0)2680 4434 y Fn(\()p Fj(x)p Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)476 4644 y Fn(+)p Fj(\024)625 4527 y Fh(Z)707 4553 y Fe(1)671 4716 y Fg(0)799 4644 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)1048 4603 y Fe(0)1071 4644 y Fn(\()p Fj(x)p Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx:)-94 4849 y Fn(Hence,)34 b(from)d(\(4.2\))h(w)m(e)h(get)1110 5060 y Fj(h)1166 5019 y Fg(2)1233 5060 y Ff(\024)c Fj(\024)1395 5019 y Fe(\000)p Fg(1)1489 5060 y Fj(\036)1547 5075 y Fg(0)1586 5060 y Fn(\()p Fj(f)11 b Fn(\(0\)\))22 b(+)g Fj(I)2009 5075 y Fg(1)2070 5060 y Fn(+)g Fj(I)2211 5075 y Fg(2)2273 5060 y Fn(+)g Fj(I)2414 5075 y Fg(3)2453 5060 y Fj(;)-94 5271 y Fn(where)33 b Fj(\036)245 5286 y Fg(0)317 5271 y Fn(is)f(de\014ned)i(in)e(\(3.2\),)-94 5526 y Fj(I)-51 5541 y Fg(1)16 5526 y Fn(:=)27 b Ff(\000)p Fn(2)289 5409 y Fh(Z)372 5435 y Fi(\024)413 5412 y Fa(\000)p Fb(\032)335 5598 y Fg(0)518 5526 y Fj(A)591 5485 y Fe(0)614 5526 y Fn(\()p Fj(t)p Fn(\))p Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(B)5 b Fn(\()p Fj(t)p Fn(\))p Fj(dt;)2287 b Fn(\(4.5\))-94 5823 y Fj(I)-51 5838 y Fg(2)16 5823 y Fn(:=)146 5736 y Ff(p)p 229 5736 V 87 x Fn(2)295 5706 y Fh(Z)378 5732 y Fe(1)341 5895 y Fg(0)469 5823 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b(+)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1056 5782 y Fg(2)1095 5823 y Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)23 b Fn(+)f(6)p Fj(\024)1662 5782 y Fe(\000)p Fg(1)1773 5706 y Fh(Z)1856 5732 y Fe(1)1819 5895 y Fg(0)1947 5823 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)2196 5782 y Fe(0)2219 5823 y Fn(\()p Fj(x)p Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx;)805 b Fn(\(4.6\))p eop %%Page: 20 20 20 19 bop 94 213 a Fn(20)94 531 y Fj(I)137 546 y Fg(3)205 531 y Fn(:=)27 b Ff(\000)412 444 y(p)p 496 444 49 4 v 496 531 a Fn(2)561 414 y Fh(Z)644 440 y Fe(1)607 603 y Fg(0)736 531 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)21 b(+)h Fj(f)11 b Fn(\()p Fj(x)p Fn(\))1322 490 y Fg(2)1362 531 y Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx)22 b Ff(\000)h Fn(6)p Fj(\024)1935 490 y Fe(\000)p Fg(1)2046 414 y Fh(Z)2129 440 y Fe(1)2092 603 y Fg(0)2220 531 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)2469 490 y Fe(0)2492 531 y Fn(\()p Fj(x)p Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx:)715 b Fn(\(4.7\))94 818 y(F)-8 b(or)32 b(all)f Fj(f)11 b Fn(\(0\))27 b Ff(2)p Fn(]0)p Fj(;)17 b Fn(1],)32 b(w)m(e)i(ha)m(v)m(e)g Fj(\036)1364 833 y Fg(0)1403 818 y Fn(\()p Fj(f)11 b Fn(\(0\)\))27 b Ff(\024)h Fn(2)1844 781 y Fe(\000)1909 754 y Fd(3)p 1909 766 31 4 v 1909 807 a(2)1953 818 y Fn(.)44 b(Then,)33 b(w)m(e)h(get)94 1036 y Fj(h)150 995 y Fg(2)218 1036 y Ff(\024)28 b Fj(\024)379 995 y Fe(\000)p Fg(1)473 1036 y Fn(2)522 995 y Fe(\000)587 968 y Fd(3)p 587 980 V 587 1021 a(2)654 1036 y Fn(+)22 b Fj(I)795 1051 y Fg(1)856 1036 y Fn(+)g Fj(I)997 1051 y Fg(2)1059 1036 y Fn(+)g Fj(I)1200 1051 y Fg(3)1239 1036 y Fj(:)2407 b Fn(\(4.8\))94 1241 y(F)-8 b(or)26 b Fj(\032)i Ff(2)p Fn(]0)p Fj(;)537 1202 y Fg(1)p 537 1218 36 4 v 537 1275 a(8)583 1241 y Fn([,)g(w)m(e)f(cut)h(the)f(in)m (tegration)e(in)m(terv)-5 b(al)25 b(in)h([0)p Fj(;)17 b(\024)2247 1205 y Fe(\000)p Fi(\032)2342 1241 y Fn(])27 b(and)g([)p Fj(\024)2663 1205 y Fe(\000)p Fi(\032)2758 1241 y Fj(;)17 b Fn(+)p Ff(1)p Fn([.)41 b(As)27 b(the)g(functions)g Fj(f)11 b Fn(,)94 1361 y Fj(f)153 1325 y Fe(0)199 1361 y Fn(are)22 b(b)s(ounded)h(and)f(the)h(function)f Fj(S)28 b Fn(admits)21 b(for)h(upp)s(er)h(b)s(ound)f(and)g(lo)m(w)m(er)h(b)s (ound)f(an)h(exp)s(onen)m(tial)94 1482 y(p)s(olynomial)h(in)i(the)i (form)d(of)i Fj(P)14 b Fn(\()p Fj(x)p Fn(\))j(exp\()p Ff(\000)1682 1400 y(p)p 1766 1400 49 4 v 1766 1482 a Fn(2)o Fj(x)p Fn(\),)29 b(the)e(in)m(tegral)f(in)g(\(4.6\))g(on)h(the)h (in)m(terv)-5 b(al)25 b([)p Fj(\024)3532 1445 y Fe(\000)p Fi(\032)3628 1482 y Fj(;)17 b Fn(+)p Ff(1)p Fn([)94 1602 y(are)33 b(equal)f(to)h Ff(O)s Fn(\()p Fj(\024)810 1566 y Fi(n)857 1602 y Fn(\))f(for)g(all)f Fj(n)d Ff(2)g Fj(I)-25 b(N)10 b Fn(.)43 b(According)33 b(to)f(\(2.6\),)g(\(2.27\))g(and)h (\(3.50\),)f(w)m(e)h(get)208 1882 y Ff(\000)p Fn(2)351 1765 y Fh(Z)434 1791 y Fi(\024)475 1768 y Fa(\000)p Fb(\032)397 1954 y Fg(0)579 1882 y Fj(A)652 1841 y Fe(0)676 1882 y Fn(\()p Fj(t)p Fn(\))p Fj(f)11 b Fn(\()p Fj(t)p Fn(\))p Fj(B)5 b Fn(\()p Fj(t)p Fn(\))p Fj(dt)208 2100 y Ff(\024)28 b(\000)p Fn(\()p Fj(f)11 b Fn(\(0\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)908 2064 y Fg(1)p Fe(\000)p Fi(\032)1039 2100 y Fn(\)\))1132 1983 y Fh(Z)1214 2009 y Fi(\024)1255 1986 y Fa(\000)p Fb(\032)1177 2171 y Fg(0)1360 2100 y Fn(exp)q(\()p Ff(\000)1624 2013 y(p)p 1707 2013 V 87 x Fn(2)p Fj(x)p Fn(\))1866 1979 y Fh(\022)1927 2100 y Fj(\024)1983 2059 y Fg(2)2023 2100 y Fj(h)2079 2059 y Fg(4)2135 2100 y Fn(exp)q(\()p Ff(\000)2399 2013 y(p)p 2482 2013 V 87 x Fn(2)p Fj(x)p Fn(\))h Ff(\000)2756 2032 y Fn(1)p 2756 2077 V 2756 2168 a(8)2837 2100 y(+)f(3)p Fj(\024)3040 2059 y Fg(2)3079 2100 y Fj(h)3135 2059 y Fg(4)3197 2100 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)3471 2059 y Fi(\032)3512 2100 y Fn(\))3550 1979 y Fh(\023)3627 2100 y Fj(dx:)94 2381 y Fn(F)-8 b(rom)32 b(\(3.5\))g(and)g(as)h Fj(e)938 2345 y Fe(\000)993 2290 y(p)p 1052 2290 36 3 v 55 x Fg(2)p Fi(\024)1128 2321 y Fa(\000)p Fb(\032)1244 2381 y Fn(=)27 b Ff(O)s Fn(\()p Fj(\024)1523 2345 y Fi(n)1570 2381 y Fn(\))33 b(for)f Fj(\024)h Fn(small,)d(it)h(results)i(that)94 2631 y Fj(I)137 2646 y Fg(1)205 2631 y Ff(\024)28 b(\000)422 2563 y Fn(3)p 397 2607 98 4 v 397 2699 a(32)527 2631 y(+)22 b Ff(O)s Fn(\()p Fj(\024)801 2589 y Fi(\032)841 2631 y Fn(\))p Fj(:)2767 b Fn(\(4.9\))94 2861 y Fk(Step)38 b(2:)50 b(Estimate)36 b(for)h Fj(I)1177 2876 y Fg(2)94 2981 y Fn(On)i(the)f(other)g(hand,)i(using)e(Prop)s(osition)f(3.8,)i (and)f(making)f(the)h(scaling)f Fj(u)g Fn(=)g(exp)q(\()p Ff(\000)3471 2899 y(p)p 3554 2899 49 4 v 82 x Fn(2)p Fj(x)p Fn(\),)j(w)m(e)94 3102 y(get)212 3259 y Fh(Z)295 3286 y Fi(\024)336 3262 y Fa(\000)p Fb(\032)258 3448 y Fg(0)440 3376 y Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)212 3605 y Fn(=)325 3566 y Fg(1)p 325 3582 36 4 v 325 3639 a(4)387 3488 y Fh(Z)470 3514 y Fg(1)433 3676 y Fi(e)466 3657 y Fa(\000)514 3616 y(p)p 564 3616 31 3 v 41 x Fd(2)p Fb(\024)630 3642 y Fa(\000)p Fb(\032)749 3456 y Ff(p)p 832 3456 322 4 v 81 x Fn(2)p Fj(u)22 b Ff(\000)g Fj(u)1114 3509 y Fg(2)p 749 3582 405 4 v 923 3673 a Fj(u)1163 3605 y(du)g Fn(+)g Ff(O)s Fn(\()p Fj(\024)1566 3564 y Fi(\032)1606 3605 y Fn(\))28 b(=)1775 3484 y Fh(\024)1829 3537 y Fn(1)p 1829 3582 49 4 v 1829 3673 a(4)1888 3518 y Ff(p)p 1971 3518 322 4 v 87 x Fn(2)p Fj(u)21 b Ff(\000)i Fj(u)2253 3576 y Fg(2)2314 3605 y Fn(+)2422 3537 y(1)p 2422 3582 49 4 v 2422 3673 a(4)2497 3605 y(arcsin\()p Fj(u)f Ff(\000)g Fn(1\))3049 3484 y Fh(\025)3093 3500 y Fg(1)3093 3684 y Fi(e)3126 3665 y Fa(\000)3174 3623 y(p)p 3223 3623 31 3 v 3223 3665 a Fd(2)q Fb(\024)3290 3650 y Fa(\000)p Fb(\032)3404 3605 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)3678 3564 y Fi(\032)3719 3605 y Fn(\))94 3862 y(hence,)35 b(for)d Fj(\024)g Fn(small)136 4003 y Fh(Z)219 4030 y Fi(\024)260 4006 y Fa(\000)p Fb(\032)182 4192 y Fg(0)364 4120 y Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Fn(=)809 4053 y Fj(\031)p 809 4097 59 4 v 814 4189 a Fn(8)900 4120 y(+)1008 4053 y(1)p 1008 4097 49 4 v 1008 4189 a(4)1089 4120 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1363 4079 y Fi(\032)1403 4120 y Fn(\))p Fj(:)3625 4108 y Fn(\(4.10\))94 4360 y(F)-8 b(rom)32 b(\(2.6\),)g(\(4.10\))f(and)i(Prop)s(osition)e(3.3)h(\(see)i (\(3.30\)\),)e(it)f(results)i(that)136 4640 y(6)p Fj(\024)241 4604 y Fe(\000)p Fg(1)352 4523 y Fh(Z)435 4549 y Fi(\024)476 4526 y Fa(\000)p Fb(\032)398 4711 y Fg(0)580 4640 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)829 4599 y Fe(0)852 4640 y Fn(\()p Fj(x)p Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Ff(\024)h Fn(6)p Fj(\024)1525 4599 y Fe(\000)p Fg(1)1619 4640 y Fn(\()p Fj(f)11 b Fn(\(0\))21 b(+)h Ff(O)s Fn(\()p Fj(\024)2136 4599 y Fg(1)p Fe(\000)p Fi(\032)2267 4640 y Fn(\)\))2360 4523 y Fh(Z)2442 4549 y Fi(\024)2483 4526 y Fa(\000)p Fb(\032)2406 4711 y Fg(0)2588 4640 y Fj(\024)2644 4599 y Fg(2)2683 4640 y Fj(h)2739 4599 y Fg(2)2779 4640 y Fn(\(1)g Ff(\000)h Fn(exp)q(\()p Ff(\000)3252 4553 y(p)p 3335 4553 V 87 x Fn(2)p Fj(x)p Fn(\)\))p Fj(:S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx:)94 4911 y Fn(F)-8 b(rom)32 b(\(3.5\))g(and)g(making)f(the)i(scaling)f Fj(u)27 b Fn(=)g(exp)q(\()p Ff(\000)2056 4829 y(p)p 2140 4829 V 2140 4911 a Fn(2)p Fj(x)p Fn(\),)33 b(w)m(e)g(get)458 5074 y Fh(Z)541 5100 y Fi(\024)582 5076 y Fa(\000)p Fb(\032)504 5262 y Fg(0)669 5191 y Fn(\(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)1142 5104 y(p)p 1225 5104 V 87 x Fn(2)p Fj(x)p Fn(\)\))p Fj(:S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Fn(=)1877 5123 y(1)p 1877 5168 V 1877 5259 a(4)1952 5074 y Fh(Z)2035 5100 y Fg(1)1998 5262 y Fi(e)2031 5243 y Fa(\000)2079 5202 y(p)p 2129 5202 31 3 v 41 x Fd(2)p Fb(\024)2195 5228 y Fa(\000)p Fb(\032)2314 5123 y Fn(\(1)22 b Ff(\000)h Fj(u)p Fn(\))p 2314 5168 302 4 v 2437 5259 a Fj(u)2626 5104 y Ff(p)p 2709 5104 322 4 v 87 x Fn(2)p Fj(u)f Ff(\000)g Fj(u)2991 5162 y Fg(2)3030 5191 y Fj(du)g Fn(+)g Ff(O)s Fn(\()p Fj(\024)3433 5150 y Fi(\032)3473 5191 y Fn(\))458 5410 y(=)561 5289 y Fh(\024)605 5410 y Fn(\()653 5343 y(3)p 653 5387 49 4 v 653 5479 a(8)734 5410 y Ff(\000)843 5343 y Fj(u)p 843 5387 56 4 v 846 5479 a Fn(8)909 5410 y(\))947 5324 y Ff(p)p 1030 5324 322 4 v 86 x Fn(2)p Fj(u)f Ff(\000)i Fj(u)1312 5382 y Fg(2)1373 5410 y Fn(+)1481 5343 y(1)p 1481 5387 49 4 v 1481 5479 a(8)1556 5410 y(arcsin\()p Fj(u)f Ff(\000)g Fn(1\))2108 5289 y Fh(\025)2152 5305 y Fg(1)2152 5489 y Fi(e)2185 5470 y Fa(\000)2233 5429 y(p)p 2282 5429 31 3 v 2282 5470 a Fd(2)q Fb(\024)2349 5455 y Fa(\000)p Fb(\032)2463 5410 y Fn(+)g Ff(O)s Fn(\()p Fj(\024)2737 5374 y Fi(\032)2778 5410 y Fn(\))p Fj(;)94 5668 y Fn(hence)926 5706 y Fh(Z)1009 5732 y Fi(\024)1050 5709 y Fa(\000)p Fb(\032)972 5895 y Fg(0)1137 5823 y Fn(\(1)g Ff(\000)h Fn(exp)q(\()p Ff(\000)1610 5736 y(p)p 1693 5736 49 4 v 87 x Fn(2)p Fj(x)p Fn(\)\))p Fj(:S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Fn(=)2364 5756 y Fj(\031)p 2345 5800 98 4 v 2345 5892 a Fn(16)2475 5823 y(+)2583 5756 y(1)p 2583 5800 49 4 v 2583 5892 a(4)2663 5823 y(+)22 b Ff(O)s Fn(\()p Fj(\024)2937 5782 y Fi(\032)2978 5823 y Fn(\))p Fj(:)p eop %%Page: 21 21 21 20 bop 3587 213 a Fn(21)-94 511 y(F)-8 b(rom)31 b(\(3.3\))h(and)g (\(3.5\),)g(w)m(e)i(get)-94 757 y(6)p Fj(\024)11 716 y Fe(\000)p Fg(1)121 640 y Fh(Z)204 666 y Fi(\024)245 642 y Fa(\000)p Fb(\032)167 828 y Fg(0)350 757 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)599 716 y Fe(0)622 757 y Fn(\()p Fj(x)p Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Ff(\024)1199 689 y Fn(3)p 1199 733 49 4 v 1199 825 a(2)1258 757 y(\()1325 689 y Fj(\031)p 1306 733 98 4 v 1306 825 a Fn(16)1435 757 y(+)1543 689 y(1)p 1543 733 49 4 v 1543 825 a(4)1602 757 y(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)1936 716 y Fi(\032)1977 757 y Fn(\))p Fj(:)1394 b Fn(\(4.11\))-94 977 y(According)32 b(to)g(\(2.6\))g(and)h (\(4.10\),)f(w)m(e)h(deduce)h(that)-94 1149 y Ff(p)p -11 1149 V 87 x Fn(2)54 1119 y Fh(Z)137 1145 y Fi(\024)178 1122 y Fa(\000)p Fb(\032)100 1307 y Fg(0)282 1236 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1+)p Fj(f)g Fn(\()p Fj(x)p Fn(\))825 1195 y Fg(2)864 1236 y Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Ff(\024)1338 1149 y(p)p 1421 1149 V 87 x Fn(2)1487 1140 y Fh(\020)1536 1236 y Fj(f)11 b Fn(\(0\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2016 1195 y Fg(1)p Fe(\000)p Fi(\032)2147 1236 y Fn(\))2185 1140 y Fh(\021)16 b(\020)2300 1236 y Fn(1)22 b(+)g(\()p Fj(f)11 b Fn(\(0\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2987 1195 y Fg(1)p Fe(\000)p Fi(\032)3117 1236 y Fn(\)\))3193 1195 y Fg(2)3233 1140 y Fh(\021)3299 1119 y(Z)3382 1145 y Fi(\024)3423 1122 y Fa(\000)p Fb(\032)3345 1307 y Fg(0)3527 1236 y Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx:)-94 1470 y Fn(F)-8 b(rom)31 b(\(4.10\),)g(w)m(e)j(get)-94 1628 y Ff(p)p -11 1628 V 87 x Fn(2)54 1598 y Fh(Z)137 1624 y Fi(\024)178 1601 y Fa(\000)p Fb(\032)100 1787 y Fg(0)282 1715 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b(+)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))869 1674 y Fg(2)909 1715 y Fn(\))p Fj(S)6 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)27 b Ff(\024)1393 1648 y Fn(3)p 1393 1692 V 1393 1784 a(2)1451 1715 y(\()1499 1648 y Fj(\031)p 1499 1692 59 4 v 1504 1784 a Fn(8)1590 1715 y(+)1698 1648 y(1)p 1698 1692 49 4 v 1698 1784 a(4)1757 1715 y(\))22 b(+)g Ff(O)s Fn(\()p Fj(\024)2091 1674 y Fi(\032)2132 1715 y Fn(\))p Fj(:)1239 b Fn(\(4.12\))-94 1936 y(According)32 b(to)g(\(4.6\),)g(\(4.11\))g(and)h(\(4.12\),)f(w)m (e)h(get)-94 2166 y Fj(I)-51 2181 y Fg(2)16 2166 y Ff(\024)131 2099 y Fn(9)p Fj(\031)p 131 2143 108 4 v 136 2235 a Fn(32)271 2166 y(+)379 2099 y(3)p 379 2143 49 4 v 379 2235 a(4)459 2166 y(+)22 b Ff(O)s Fn(\()p Fj(\024)733 2125 y Fi(\032)774 2166 y Fn(\))p Fj(:)2597 b Fn(\(4.13\))-94 2378 y Fk(Step)37 b(3:)50 b(Estimate)36 b(for)h Fj(I)988 2393 y Fg(3)-94 2498 y Fn(F)-8 b(rom)31 b(Prop)s(osition)f(3.3)j(\(see)g(\(3.30\)\))f (and)h(\(3.5\),)f(on)g(the)h(in)m(terv)-5 b(al)32 b([0)p Fj(;)17 b(\024)2641 2462 y Fe(\000)p Fi(\032)2735 2498 y Fn(],)33 b(w)m(e)h(ha)m(v)m(e)-94 2733 y Fj(T)41 b Ff(\025)28 b Fj(U)38 b Fn(:=)354 2666 y Fj(B)p 354 2710 80 4 v 355 2801 a(C)443 2733 y(;)2966 b Fn(\(4.14\))-94 2945 y(where)-94 3176 y Fj(B)5 b Fn(\()p Fj(x)p Fn(\))28 b(:=)284 3027 y Ff(p)p 367 3027 49 4 v 82 x Fn(2)p 284 3153 132 4 v 326 3245 a(2)443 3080 y Fh(\020)492 3176 y Fn(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)927 3089 y(p)p 1010 3089 49 4 v 87 x Fn(2)p Fj(x)p Fn(\))f(+)g Ff(O)s Fn(\()p Fj(\024)1448 3135 y Fi(\032)1489 3176 y Fn(\))1527 3080 y Fh(\021)1593 3176 y Fj(:S;)1729 b Fn(\(4.15\))-94 3388 y(and)-94 3630 y Fj(C)7 b Fn(\()p Fj(x)p Fn(\))27 b(:=)h Fj(S)6 b Fn(+)424 3481 y Ff(p)p 507 3481 V 82 x Fn(2)p 424 3607 132 4 v 465 3699 a(4)582 3534 y Fh(\020)632 3630 y Fn(1)22 b Ff(\000)g Fn(exp)q(\()p Ff(\000)1066 3543 y(p)p 1150 3543 49 4 v 1150 3630 a Fn(2)o Fj(x)p Fn(\))h(+)f Ff(O)s Fn(\()p Fj(\024)1588 3589 y Fi(\032)1628 3630 y Fn(\))1666 3534 y Fh(\021)1733 3630 y Fj(:S)6 b Fn(+)1902 3469 y Fh(s)p 1984 3469 1484 4 v 1984 3630 a Fj(S)2050 3601 y Fg(2)2112 3630 y Fn(+)2220 3563 y(1)p 2220 3607 49 4 v 2220 3699 a(8)2295 3534 y Fh(\020)2345 3630 y Fn(1)22 b Ff(\000)g Fn(exp)q(\()p Ff(\000)2779 3543 y(p)p 2863 3543 V 2863 3630 a Fn(2)o Fj(x)p Fn(\))h(+)f Ff(O)s Fn(\()p Fj(\024)3301 3601 y Fi(\032)3341 3630 y Fn(\))3379 3534 y Fh(\021)3429 3557 y Fg(2)3468 3630 y Fj(:)p Fn(\(4.16\))-94 3865 y(As)34 b(the)h(functions)g Fj(f)703 3829 y Fe(0)760 3865 y Fn(and)f Fj(S)40 b Fn(are)35 b(p)s(ositiv)m(e)e(on)i Fj(I)-25 b(R)1809 3829 y Fg(+)1867 3865 y Fn(,)35 b(from)e(\(4.4\),)i(w)m(e)g (deduce)h(that)e Fj(T)45 b Ff(\025)31 b Fn(0.)48 b(Th)m(us,)-94 3985 y(from)31 b(\(2.6\))h(and)h(\(4.14\),)e(w)m(e)j(get)-94 4230 y Ff(\000)-17 4144 y(p)p 66 4144 V 86 x Fn(2)131 4113 y Fh(Z)214 4140 y Fe(1)178 4302 y Fg(0)306 4230 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\)\(1)22 b(+)g Fj(f)11 b Fn(\()p Fj(x)p Fn(\))893 4189 y Fg(2)932 4230 y Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Ff(\024)g(\000)1498 4163 y Fn(3)p 1498 4207 V 1498 4299 a(2)1574 4113 y Fh(Z)1657 4140 y Fi(\024)1698 4116 y Fa(\000)p Fb(\032)1620 4302 y Fg(0)1802 4230 y Fj(U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)23 b Fn(+)f Ff(O)s Fn(\()p Fj(\024)2412 4189 y Fi(\032)2453 4230 y Fn(\))p Fj(:)918 b Fn(\(4.17\))-94 4445 y(Moreo)m(v)m(er,)34 b(w)m(e)f(ha)m(v)m(e)-94 4685 y Ff(\000)p Fn(6)p Fj(\024)88 4644 y Fe(\000)p Fg(1)199 4568 y Fh(Z)282 4594 y Fe(1)245 4756 y Fg(0)373 4685 y Fj(f)11 b Fn(\()p Fj(x)p Fn(\))p Fj(f)622 4644 y Fe(0)645 4685 y Fn(\()p Fj(x)p Fn(\))p Fj(T)j Fn(\()p Fj(x)p Fn(\))p Fj(dx)28 b Ff(\024)h(\000)1305 4618 y Fn(3)p 1305 4662 V 1305 4753 a(2)1380 4568 y Fh(Z)1463 4594 y Fi(\024)1504 4571 y Fa(\000)p Fb(\032)1426 4756 y Fg(0)1592 4685 y Fn(\(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)2065 4598 y(p)p 2148 4598 V 87 x Fn(2)p Fj(x)p Fn(\)\))p Fj(:U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)23 b Fn(+)f Ff(O)s Fn(\()p Fj(\024)2965 4644 y Fi(\032)3006 4685 y Fn(\))p Fj(:)365 b Fn(\(4.18\))-94 4921 y(Making)32 b(the)h(scaling)e Fj(u)c Fn(=)h(exp)q(\()p Ff(\000)1197 4839 y(p)p 1280 4839 V 82 x Fn(2)p Fj(x)p Fn(\),)33 b(from)e(\(3.5\),)h (\(4.14\),)g(\(4.15\))g(and)h(\(4.16\),)f(w)m(e)h(get)-94 5092 y Fh(Z)-11 5118 y Fi(\024)30 5095 y Fa(\000)p Fb(\032)-48 5280 y Fg(0)134 5209 y Fj(U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)29 b Fn(=)631 5142 y(1)p 589 5186 132 4 v 589 5204 a Ff(p)p 672 5204 49 4 v 82 x Fn(2)748 5092 y Fh(Z)831 5118 y Fg(1)794 5280 y Fi(e)827 5262 y Fa(\000)875 5220 y(p)p 924 5220 31 3 v 924 5262 a Fd(2)q Fb(\024)991 5247 y Fa(\000)p Fb(\032)1179 5142 y Fn(1)p 1110 5186 188 4 v 1110 5204 a Ff(p)p 1193 5204 49 4 v 82 x Fn(2)o Fj(u)1340 5026 y Fh(q)p 1423 5026 358 4 v 100 x Fj(u)p Fn(\(2)21 b Ff(\000)i Fj(u)p Fn(\))o(\()p Fj(u)f Ff(\000)g Fn(1\))p 1317 5186 789 4 v 1317 5302 a Fj(u)f Ff(\000)i Fn(2)f Ff(\000)1665 5202 y Fh(q)p 1748 5202 358 4 v 100 x Fj(u)p Fn(\(2)f Ff(\000)i Fj(u)p Fn(\))2115 5209 y Fj(du)e Fn(+)h Ff(O)s Fn(\()p Fj(\024)2517 5168 y Fi(\032)2558 5209 y Fn(\))p Fj(;)813 b Fn(\(4.19\))-94 5492 y(and)-94 5638 y Fh(Z)-11 5664 y Fi(\024)30 5641 y Fa(\000)p Fb(\032)-48 5827 y Fg(0)117 5755 y Fn(\(1)19 b Ff(\000)h Fn(exp)q(\()p Ff(\000)584 5668 y(p)p 668 5668 49 4 v 668 5755 a Fn(2)o Fj(x)p Fn(\)\))p Fj(:U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)30 b Fn(=)1372 5688 y(1)p 1330 5732 132 4 v 1330 5750 a Ff(p)p 1413 5750 49 4 v 82 x Fn(2)1488 5638 y Fh(Z)1571 5664 y Fg(1)1535 5827 y Fi(e)1568 5808 y Fa(\000)1616 5766 y(p)p 1665 5766 31 3 v 1665 5808 a Fd(2)q Fb(\024)1732 5793 y Fa(\000)p Fb(\032)1841 5755 y Ff(\000)1997 5688 y Fn(1)p 1928 5732 188 4 v 1928 5750 a Ff(p)p 2011 5750 49 4 v 82 x Fn(2)p Fj(u)2138 5573 y Fh(q)p 2221 5573 358 4 v 100 x Fj(u)p Fn(\(2)22 b Ff(\000)g Fj(u)p Fn(\)\()p Fj(u)g Ff(\000)g Fn(1\))2881 5636 y Fg(2)p 2135 5732 789 4 v 2135 5849 a Fj(u)g Ff(\000)h Fn(2)e Ff(\000)2483 5749 y Fh(q)p 2566 5749 358 4 v 100 x Fj(u)p Fn(\(2)g Ff(\000)i Fj(u)p Fn(\))2934 5755 y Fj(du)18 b Fn(+)h Ff(O)s Fn(\()p Fj(\024)3330 5714 y Fi(\032)3371 5755 y Fn(\))p Fj(:)p Fn(\(4.20\))p eop %%Page: 22 22 22 21 bop 94 213 a Fn(22)94 511 y(F)-8 b(or)32 b Fj(u)27 b Ff(2)p Fn(]0)p Fj(;)17 b Fn(2[,)33 b(w)m(e)h(ha)m(v)m(e)526 685 y Fh(Z)705 734 y Fn(1)p 635 778 188 4 v 635 796 a Ff(p)p 718 796 49 4 v 82 x Fn(2)p Fj(u)866 619 y Fh(q)p 949 619 358 4 v 100 x Fj(u)p Fn(\(2)21 b Ff(\000)i Fj(u)p Fn(\))o(\()p Fj(u)f Ff(\000)g Fn(1\))p 843 778 789 4 v 843 895 a Fj(u)f Ff(\000)i Fn(2)f Ff(\000)1190 795 y Fh(q)p 1273 795 358 4 v 100 x Fj(u)p Fn(\(2)g Ff(\000)g Fj(u)p Fn(\))1641 802 y Fj(du)27 b Fn(=)1888 652 y Ff(p)p 1971 652 49 4 v 82 x Fn(2)p 1888 778 132 4 v 1930 870 a(4)2047 705 y Fh(\020)2096 802 y Fn(arcsin\()p Fj(u)22 b Ff(\000)g Fn(1\))g Ff(\000)h Fj(u)e Fn(+)2945 715 y Ff(p)p 3028 715 322 4 v 87 x Fn(2)p Fj(u)h Ff(\000)g Fj(u)3310 773 y Fg(2)3349 705 y Fh(\021)3416 802 y Fj(;)94 1086 y Fn(and)786 1167 y Fh(Z)886 1284 y Ff(\000)1042 1217 y Fn(1)p 973 1261 188 4 v 973 1279 a Ff(p)p 1056 1279 49 4 v 82 x Fn(2)p Fj(u)1184 1101 y Fh(q)p 1267 1101 358 4 v 100 x Fj(u)p Fn(\(2)f Ff(\000)i Fj(u)p Fn(\))o(\()p Fj(u)f Ff(\000)g Fn(1\))1926 1165 y Fg(2)p 1180 1261 789 4 v 1180 1377 a Fj(u)g Ff(\000)h Fn(2)f Ff(\000)1528 1277 y Fh(q)p 1611 1277 358 4 v 100 x Fj(u)p Fn(\(2)g Ff(\000)g Fj(u)p Fn(\))1979 1284 y Fj(du)786 1581 y Fn(=)900 1514 y(1)p 900 1558 49 4 v 900 1649 a(8)975 1581 y(arcsin\()p Fj(u)f Ff(\000)i Fn(1\))f(+)1657 1432 y Ff(p)p 1740 1432 V 82 x Fn(2)p 1657 1558 132 4 v 1698 1649 a(8)1799 1494 y Ff(p)p 1882 1494 322 4 v 87 x Fn(2)p Fj(u)f Ff(\000)i Fj(u)2164 1552 y Fg(2)2203 1581 y Fn(\(3)e Ff(\000)i Fj(u)p Fn(\))f Ff(\000)2636 1432 y(p)p 2719 1432 49 4 v 82 x Fn(2)p 2636 1558 132 4 v 2678 1649 a(4)2778 1581 y Fj(u)p Fn(\(1)f Ff(\000)3052 1514 y Fj(u)p 3052 1558 56 4 v 3055 1649 a Fn(2)3117 1581 y(\))p Fj(:)94 1773 y Fn(F)-8 b(rom)32 b(\(4.19\),)f(it)h(results)h(that)94 1890 y Fh(Z)178 1917 y Fi(\024)219 1893 y Fa(\000)p Fb(\032)141 2079 y Fg(0)323 2008 y Fj(U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)29 b Fn(=)778 1940 y Fj(\031)p 778 1984 59 4 v 783 2076 a Fn(8)869 2008 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1143 1966 y Fi(\032)1184 2008 y Fn(\))p Fj(:)2376 b Fn(\(4.21\))94 2218 y(Moreo)m(v)m(er,)35 b(from)c(\(4.20\),)h(w)m(e)i(get)94 2336 y Fh(Z)178 2362 y Fi(\024)219 2339 y Fa(\000)p Fb(\032)141 2524 y Fg(0)306 2453 y Fn(\(1)22 b Ff(\000)h Fn(exp)q(\()p Ff(\000)779 2366 y(p)p 862 2366 49 4 v 87 x Fn(2)p Fj(x)p Fn(\)\))p Fj(:U)10 b Fn(\()p Fj(x)p Fn(\))p Fj(dx)29 b Fn(=)1525 2385 y(1)p 1525 2430 V 1525 2521 a(8)1583 2453 y(\(1)22 b(+)1800 2385 y Fj(\031)p 1800 2430 59 4 v 1805 2521 a Fn(2)1869 2453 y(\))g(+)g Ff(O)s Fn(\()p Fj(\024)2203 2412 y Fi(\032)2244 2453 y Fn(\))p Fj(:)1316 b Fn(\(4.22\))94 2663 y(F)-8 b(rom)32 b(\(4.17\),)f(\(4.18\),)h (\(4.21\))g(and)h(\(4.22\),)f(w)m(e)h(get)94 2884 y Fj(I)137 2899 y Fg(3)205 2884 y Ff(\024)28 b(\000)397 2816 y Fn(9)p Fj(\031)p 397 2860 108 4 v 402 2952 a Fn(32)537 2884 y Ff(\000)671 2816 y Fn(3)p 647 2860 98 4 v 647 2952 a(16)776 2884 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1050 2842 y Fi(\032)1091 2884 y Fn(\))p Fj(:)2469 b Fn(\(4.23\))94 3091 y Fk(Step)38 b(4:)50 b(Upp)s(er)38 b(b)s(ound)h(for)e Fj(A)1428 3055 y Fe(0)1452 3091 y Fn(\(0\))94 3211 y(Th)m(us,)43 b(according)c(to)g(\(4.8\),)i(\(4.9\),)g(\(4.13\))d(and)i(\(4.23\),)h (w)m(e)f(deduce)h(that)e(there)h(exists)h Fj(\024)3579 3226 y Fg(0)3618 3211 y Fn(,)h(suc)m(h)94 3331 y(that,)33 b(for)f(all)e Fj(\024)e Ff(\024)g Fj(\024)862 3346 y Fg(0)935 3331 y Fn(and)k(all)f(pairs)h(\()p Fj(f)5 b(;)17 b(A)p Fn(\))32 b(solutions)g(of)g(\()p Fj(GL)p Fn(\))2515 3346 y Fe(1)2622 3331 y Fn(satisfying)g(\(3.3\),)g(w)m(e)i(ha)m(v)m(e) 136 3552 y Fj(h)192 3516 y Fg(2)259 3552 y Ff(\024)28 b Fj(\024)420 3516 y Fe(\000)p Fg(1)515 3552 y Fn(2)564 3515 y Fe(\000)629 3488 y Fd(3)p 629 3500 31 4 v 629 3541 a(2)695 3552 y Fn(+)803 3485 y(15)p 803 3529 98 4 v 803 3620 a(32)933 3552 y(+)22 b Ff(O)s Fn(\()p Fj(\024)1207 3511 y Fi(\032)1247 3552 y Fn(\))p Fj(:)3625 3545 y Fn(\(4.24\))94 3759 y(F)-8 b(rom)32 b(Lemma)f(3.1)h(and)h(\(4.24\),)e(the)i(pro)s(of)f (of)g(Theorem)h(1.3)f(follo)m(ws.)94 4000 y(According)h(to)f(\(1.6\))g (and)h(Theorem)g(1.3,)f(w)m(e)i(deduce)g(the)f(P)m(arr)g(F)-8 b(orm)m(ula.)94 4176 y Fk(Theorem)38 b(4.1)49 b Fc(\(Parr)34 b(F)-7 b(ormula\))34 b(Ther)-5 b(e)34 b(exists)h Fj(\024)2066 4191 y Fg(0)2140 4176 y Fc(such)g(that,)g(for)g(al)5 b(l)34 b Fj(\024)2942 4191 y Fg(0)3009 4176 y Ff(\024)28 b Fj(\024)p Fc(,)35 b(we)g(have)1291 4408 y Fj(\024)1364 4312 y Fh(\020)1413 4408 y Fj(h)1469 4367 y Fi(sh)1547 4408 y Fn(\()p Fj(\024)p Fn(\))1679 4312 y Fh(\021)1729 4335 y Fg(2)1796 4408 y Fn(=)27 b(2)1948 4367 y Fe(\000)2013 4340 y Fd(3)p 2013 4352 31 4 v 2013 4393 a(2)2080 4408 y 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Fd(1)p 1245 749 31 4 v 1245 790 a(2)1290 805 y Fj(h)1346 764 y Fi(sh)1424 805 y Fn(\()p Fj(\024)p Fn(\))27 b Ff(\024)1727 698 y Fi(n)1689 722 y Fh(X)1691 905 y Fi(i)p Fg(=0)1825 805 y Fj(h)1881 820 y Fi(i)1909 805 y Fj(\024)1965 764 y Fi(i)2016 805 y Fn(+)22 b Fj(C)7 b(\024)2247 764 y Fi(n)p Fg(+1)2384 805 y Fj(:)-94 1075 y Fk(Ac)m(kno)m(wledgemen)m(ts)3 1195 y Fn(The)34 b(author)f(w)m(ould)g(lik)m(e)f(to)g(thank)i(C.)f (Bolley)f(and)h(B.)g(Hel\013er)g(for)f(helpful)g(discussions)i(on)f (the)-94 1316 y(problem.)-94 1554 y Fk(REFERENCES)-94 1752 y Fn(1.)86 b(A.)38 b(Aftalion)56 b(On)37 b(the)h(minimizers)c(of)j (the)h(Ginzburg-Landau)e(energy)j(for)e(high)f(k)-5 b(appa:)54 b(the)68 1872 y(one-dimensional)30 b(case.)k Fk(8)p Fn(,)f(p.)f (331-345)42 b Fc(Eur.)35 b(J.)g(Appl.)g(Math.)d Fn(\(1997\))-94 1993 y(2.)86 b(C.)29 b(Bolley)e(and)h(P)-8 b(.)28 b(Del)f(Castillo)e(:) 55 b(Existence)29 b(and)f(uniqueness)i(for)e(the)g(half-space)f (Ginzburg-)68 2113 y(Landau)33 b(mo)s(del)e(.)i 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b(ar)n(\023)-47 b(e)30 b(\(se)-5 b(ction)31 b(analyse)f(non)h(lin)n(\023)-47 b(eair)-5 b(e\),)68 3197 y(V)e(ol.)35 b(14,)f(No.)h(5,)g(597{613)p Fn(,)c(\(1997\).)-94 3317 y(6.)86 b(C.)39 b(Bolley)d(and)i(B.)g (Hel\013er)g(:)75 b(Sup)s(erheating)38 b(in)f(a)g(semi-in\014nite)f (\014lm)g(in)h(the)h(w)m(eak)i Fj(\024)d Fn(limit:)68 3437 y(n)m(umerical)44 b(results)h(and)h(appro)m(ximate)e(mo)s(dels.)79 b Fc(Math.)47 b(Mo)-5 b(del.)46 b(Num.)g(A)n(nal.)91 b Fn(\(1997\))44 b Fk(31)68 3558 y Fn(121{165)31 b(.)-94 3678 y(7.)86 b(C.)40 b(Bolley)-8 b(,)38 b(F.)h(F)-8 b(ouc)m(her)39 b(and)g(B.)f(Hel\013er)h(:)78 b(Sup)s(erheating)39 b(\014eld)f(for)g (the)h(Ginzburg-Landau)68 3798 y(equations)j(in)f(the)h(case)g(of)f(a)g (large)g(b)s(ounded)h(in)m(terv)-5 b(al.)69 b Fc(J.)43 b(Math.)g(Phys.)g(V)-7 b(ol.)43 b(41,)h(No.)f(11)68 3919 y Fn(\(2000\))32 b(7263{7289.)-94 4039 y(8.)86 b(C.)31 b(Bolley)e(and)h(B.)g(Hel\013er)g(:)55 b(A)30 b(priori)e(estimates)i (for)f(Ginzburg-Landau)g(solutions.)42 b Fc(Nonlin-)68 4160 y(e)-5 b(ar)35 b(PDE's)f(in)h(Condense)-5 b(d)33 b(Matter)j(and)e(R)-5 b(e)g(active)35 b(Flows)c Fn(\(2002\))g(355{373.) -94 4280 y(9.)86 b(S.)39 b(Chapman)f(:)78 b(Sup)s(erheating)38 b(\014eld)g(of)g(t)m(yp)s(e)i(I)s(I)e(sup)s(erconductors.)63 b Fc(Siam)39 b(J.)i(Appl.)f(Math.)68 4400 y Fn(\(1995\))32 b(Vol.)g(55,)g(No.)g(5,)h(1233{1258.)-94 4521 y(10.)k(P)-8 b(.)33 b(Del)f(Castillo)73 b(Th)m(\022)-46 b(ese,)45 b Fc(Universit)n(\023)-47 b(e)34 b(Paris)h(Sud,)f(\(D)n(\023)-47 b(ec.)33 b(2000\).)-94 4641 y Fn(11.)k(P)-8 b(.)35 b(Del)e(Castillo)f (:)60 b(Tw)m(o)35 b(terms)f(in)g(the)g(lo)m(w)m(er)g(b)s(ound)h(for)e (the)i(sup)s(erheating)f(\014eld)g(in)f(a)h(semi-)68 4761 y(in\014nite)e(\014lm)f(in)h(the)h(w)m(eak-)p Fj(\024)g Fn(limit.)40 b Fc(Eur.)c(J.)e(Appl.)h(Math.)e Fn(\(2002\))e(Vol.)h(13,) g(No.5,)h(519{547.)-94 4882 y(12.)k(P)-8 b(.)40 b(Del)d(Castillo)g(:)79 b(Expansion)39 b(for)g(the)g(sup)s(erheating)g(\014eld)g(in)f(a)g (semi-in\014nite)f(\014lm)g(in)i(the)68 5002 y(w)m(eak-)p Fj(\024)34 b Fn(limit.)40 b Fc(Math.)35 b(Mo)-5 b(del.)34 b(Num.)i(A)n(nal.,)e(Vol.)h(36,)f(No.)h(6,)g(\(2002\))c Fn(971{993.)-94 5123 y(13.)37 b(P)-8 b(.)33 b(Del)e(Castillo)f(:)54 b(Lo)m(w)m(er)33 b(b)s(ound)f(for)g(the)g(sup)s(erheating)h(\014eld)f (in)f(a)h(semi-in\014nite)e(\014lm)g(in)i(the)68 5243 y(w)m(eak-)p Fj(\024)i Fn(limit:)40 b(the)33 b(general)f(case.)44 b Fc(J.)35 b(Math.)g(Phys.)g(V)-7 b(ol.)35 b(44)f(,)h(No.)g(6,)f (\(2003\))e Fn(2416{2450.)-94 5363 y(14.)37 b(P)-8 b(.)48 b(Del)e(Castillo)f(:)113 b(Existence)49 b(and)f(Lo)s(calization)c(of)j (solutions)f(of)h(the)h(one)f(dimensional)68 5484 y(Ginzburg-Landau)32 b(system.)44 b Fc(Submitte)-5 b(d)-94 5604 y Fn(15.)37 b(S.)26 b(Di)d(Bartolo,)i(T.)h(Dorsey)g(&)f(J.)g(Dolgert.)39 b(Sup)s(erheating)25 b(\014elds)g(of)g(sup)s(erconductors)i(Asymp-)68 5725 y(totic)22 b(analysis)g(and)h(n)m(umerical)e(results.)40 b Fc(Physic)-5 b(al)25 b(R)-5 b(eview)25 b(B,)g(Vol.)h(53,)h(No)f(9)p Fn(,)f(1996.)39 b(Erratum.)68 5845 y Fc(Physic)-5 b(al)35 b(R)-5 b(eview)34 b(B,)g(Vol.)h(56,)g(No.)g(5)p Fn(,)d(1997.)p eop %%Page: 24 24 24 23 bop 94 213 a Fn(24)94 511 y(16.)38 b(P)-8 b(.)47 b(G.)f(De)g(Gennes)i(:)109 b(Sup)s(erconductivit)m(y:)73 b(selected)47 b(topics)f(in)g(solid)f(state)i(ph)m(ysics)h(and)257 632 y(theoretical)32 b(Ph)m(ysics.)45 b Fc(Pr)-5 b(o)g(c.)34 b(of)h(8th)g(L)-5 b(atin)35 b(americ)-5 b(an)33 b(scho)-5 b(ol)34 b(of)h(physics,)f(Car)-5 b(ac)g(as)40 b Fn(\(1966\).)94 752 y(17.)e(V.)28 b(L.)g(Ginzburg)54 b(:)h(On)27 b(the)h(theory)g(of)f (sup)s(erconductivit)m(y)-8 b(,)44 b Fc(Nuovo)30 b(Cimento)d Fk(2)g Fn(\(1955\))g(1234.)94 873 y(18.)38 b(V.)27 b(L.)f(Ginzburg)f(:) 55 b(On)26 b(the)h(destruction)f(and)h(the)f(onset)h(of)f(sup)s (erconductivit)m(y)h(in)e(a)h(magnetic)257 993 y(\014eld,)52 b Fc(Zh.)37 b(Eksp)-5 b(erim.)36 b(i)h(T)-7 b(e)i(or.)36 b(Fiz.)e Fk(34)i Fn(\(1958\))e(113{125.)16 b(T)-8 b(ransl.)33 b Fc(Soviet)k(Physics)g(JETP)72 b Fk(7)257 1113 y Fn(\(1958\))32 b(78{87.)94 1234 y(19.)38 b(H.)47 b(P)m(arr)f(:)107 b(Sup)s (erconductiv)m(e)48 b(sup)s(erheating)e(\014eld)g(for)f(\014nite)h Fj(\024)p Fn(.)83 b Fc(Z.)47 b(Physik)g(B25)e Fn(\(1976\))257 1354 y(359{361.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0311071306432--