一道关于方程根中点范围的问题

mathe · 发表于 2019-9-7 21:29:22

8#竟然犯了两处错误
展开式说明$r_2^2-r_1^2=2(\sum_{t=3}^{\infty}{r_1^t-(-r_2)^t}/t)$
所以$r_2-r_1=2(\sum_{t=3}^{\infty}{r_1^t-(-r_2)^t}/{t(r_1-(-r_2))})=2(\sum_{t=3}^{\infty}{r_1^{t-1}-r_1^{t-2}r_2+...+(-r_2)^{t-1}}/t)<2(\sum_{t=1}^{\infty}{r_2^{2t}}/{2t+1})={\ln({1+r_2}/{1-r_2})-2r_2}/r_2$
由于$x_2+x_1=2+r_2-r_1$,而$m+\sqrt(m)=1+r_2-\ln(1+r_2)+\sqrt{1+r_2-\ln(1+r_2)}$,估算余项可知前者约$2/3r_2^2$,后者约$3/4r_2^2$,所以在$r_2$很小时不等式必然成立。
数值计算在$r_2\lt 0.16141968259452506460954163749391989939$不等式都成立，也就是在$m\lt 1.0117765614820538214429398700588724925$时不等式都成立。

mathe · 发表于 2019-9-8 07:06:52

现在我们分析中间段的，下面表格利用数值计算（计算精度25位）证明了$1.008\le m \le 2.02$范围内不等式都成立
注意每行对应一小段区间，其中$x_1$最大值加上 $x_2$最大值正好等于$m+\sqrt{m}$的最小值，所以验证了这个区间中不等式成立。
而其中每个区间$x_1$最大值在区间左端点取到，$x_2$最大值在区间右端点取到。
这个表格中结果，结合13#/14#的结果证明了不等式总是成立的
[

m区间	$x_1$最大值	$x_2$最大值	$m+\sqrt{m}$最小值
[1.0080000000000000000000000000000000000,1.0081553313288650149075894243291361436]	0.87878505326751967053462891887147107423	1.1332069785733709870391775120707539131	2.0119920318408906575738064309422249873
[1.0081553313288650149075894243291361436,1.0083150764574697443894215052982505391]	0.87766482639292484077893715061995591643	1.1345598906503264560075891883091230560	2.0122247170432512967865263389290789725
[1.0083150764574697443894215052982505391,1.0084793996245196051232382633426262755]	0.87652482989385091029540619031516347727	1.1359391779764196436280230731142951848	2.0124640078702705539234292634294586620
[1.0084793996245196051232382633426262755,1.0086484725983911009044274038828966256]	0.87536455838705738618389941340955073776	1.1373455914261425123318873305626161596	2.0127101498131998985157867439721668974
[1.0086484725983911009044274038828966256,1.0088224750862724683714753044146556680]	0.87418348980095405242260512922414977492	1.1387799097991176826204128381702004793	2.0129633996000717350430179673943502542
[1.0088224750862724683714753044146556680,1.0090015951689498980112691528347531560]	0.87298108470896710601566055070643688530	1.1402429410960766385724051984863204847	2.0132240258050437445880657491927573700
[1.0090015951689498980112691528347531560,1.0091860297630566690011892122919343281]	0.87175678563213194994087210199029388443	1.1417355238637100202177214142641004207	2.0134923094958419701585935162543943052
[1.0091860297630566690011892122919343281,1.0093759851127458424411463692944245644]	0.87051001630933300974608480973985839656	1.1432585286126610482738293196704297326	2.0137685449219940580199141294102881292
[1.0093759851127458424411463692944245644,1.0095716773129028755344755616522282202]	0.86924018093352441100137642899503042171	1.1448128593132310697026781074153780266	2.0140530402467554807040545364104084483
[1.0095716773129028755344755616522282202,1.0097733328661838169144404977339771883]	0.86794666335217406768392122273193747424	1.1463994549736871363428630138853387679	2.0143461183258612040267842366172762421
[1.0097733328661838169144404977339771883,1.0099811892763489063042529393313761358]	0.86662882623007746033322162473180766897	1.1480192913064073641867281863580792115	2.0146481175364848245199498110898868804
[1.0099811892763489063042529393313761358,1.0101954956805618465817136023530136862]	0.86528601017258589772574244246899261526	1.1496733824874726725566753213491987712	2.0149593926600585702824177638181913865
[1.0101954956805618465817136023530136862,1.0104165135235433046139630393244945238]	0.86391753280718712563933029032571922280	1.1513627830157156661381592554316465436	2.0152803158229027917774895457573657664
[1.0104165135235433046139630393244945238,1.0106445172767050512105146211551080073]	0.86252268782126354028666920141981156298	1.1530885896776714032621328716912642414	2.0156112774989349435488020731110758044
[1.0106445172767050512105146211551080073,1.0108797952056504708329082353414035902]	0.86110074395373475447568290151036196603	1.1548519436253433248576834283993486906	2.0159526875790780793333663299097106566
[1.0108797952056504708329082353414035902,1.0111226501897100568635897619588883199]	0.85965094393816662983059493223297699472	1.1566540325742036852006610529229127279	2.0163049765123703150312559851558897227
[1.0111226501897100568635897619588883199,1.0113734005974892766567998041910128635]	0.85817250339479791654521824167881242326	1.1584960931293946885438263454400465614	2.0166685965241926050890445871188589847
[1.0113734005974892766567998041910128635,1.0116323812227434048678347818452000670]	0.85666460966879813540219507247751312832	1.1603794132486877576787851974664750568	2.0170440229174858930809802699439881851
[1.0116323812227434048678347818452000670,1.0118999442852624177335759715052913017]	0.85512642061192611699906938330939603996	1.1623053348513978356997465410288123105	2.0174317554633239526988159243382083504
[1.0118999442852624177335759715052913017,1.0121764605018519509705047422948333362]	0.85355706330460752807249311644184737650	1.1642752565831416086965377607131324150	2.0178323198877491367690308771549797915
[1.0121764605018519509705047422948333362,1.0124623202329371215137716535685031606]	0.85195563271529164594039268671054760927	1.1662906367470776919348938616140135512	2.0182462694623693378752865483245611604
[1.0124623202329371215137716535685031606,1.0127579347107985439758985692685868916]	0.85032119029378251357132292484222614331	1.1683529964130782404874300875944264844	2.0186741867068607540587530124366526277
[1.0127579347107985439758985692685868916,1.0130637373559783981915549025056639601]	0.84865276249506739738175004850820267503	1.1704639227171607032629144896395496779	2.0191166852122281006446645381477523530
[1.0130637373559783981915549025056639601,1.0133801851889736498850150220056694398]	0.84694933922998622165249989476144058473	1.1726250723644616406665100620283413732	2.0195744115944478623190099567897819579
[1.0133801851889736498850150220056694398,1.0137077603449687343223180423141649773]	0.84520987223889949301561039752811473127	1.1748381753500683536271330760121528488	2.0200480475889678466427434735402675801
[1.0137077603449687343223180423141649773,1.0140469717000570006624905655248361843]	0.84343327338431938066403561480103576314	1.1771050389131458440111404618039722614	2.0205383122974652246751760766050080246
[1.0140469717000570006624905655248361843,1.0143983566181654426607438271595963504]	0.84161841285826942794051651729416477334	1.1794275517410143595616384077729742204	2.0210459645992837875021549250671389937
[1.0143983566181654426607438271595963504,1.0147624828287378857845957740418149130]	0.83976411729993332889390728401662521321	1.1818076884411552517285649848710047296	2.0215718057410885806224722688876299428
[1.0147624828287378857845957740418149130,1.0151399504461558373013179842195034964]	0.83786916781894297330968495968531617401	1.1842475143005597126224563377345041468	2.0221166821195026859321412974198203208
[1.0151399504461558373013179842195034964,1.0155313941428925020602304162497640887]	0.83593229791944141758789614856271609667	1.1867491903533966985294436877221542118	2.0226814882728381161173398362848703084
[1.0155313941428925020602304162497640887,1.0159374854895138857396733222597635149]	0.83395219131983869631271112050250198729	1.1893149787796745410514272993290414960	2.0232671700995132373641384198315434833
[1.0159374854895138857396733222597635149,1.0163589354758724240059413025554707128]	0.83192747966295886434655242518705635096	1.1919472486594180465327418932453252998	2.0238747283223769108792943184323816508
[1.0163589354758724240059413025554707128,1.0167964972291954061533933298345248429]	0.82985674011105711439302961107776854603	1.1946484821088931524252441203803944824	2.0245052222199502668182737314581630284
[1.0167964972291954061533933298345248429,1.0172509689462662069331047652845044565]	0.82773849281996842508111547603458254846	1.1974212808275996301265645434105145013	2.0251597736475680552076800194450970498
[1.0172509689462662069331047652845044565,1.0177231970585461493164699584133091210]	0.82557119828643662106870295521023889535	1.2002683730871355265825948550078848624	2.0258395713735721476512978102181237578
[1.0177231970585461493164699584133091210,1.0182140796509055703374624983585138635]	0.82335325456246820376825397751040249483	1.2031926211956332269705733233928796465	2.0265458757581014307388273009032821414
[1.0182140796509055703374624983585138635,1.0187245701566431569114103404609561573]	0.82108299433036274715903259597312521442	1.2061970294742961215425566484965752551	2.0272800238046588687015892444697004695
[1.0187245701566431569114103404609561573,1.0192556813536938160160964531779392004]	0.81875868183189574698453512145808582455	1.2092847527856486641706383546423733369	2.0280434346175444111551734761004591615
[1.0192556813536938160160964531779392004,1.0198084896893806026128701504490923825]	0.81637850964497619025895942928648922736	1.2124591056564749417143019007849085780	2.0288376153014511319732613300713978054
[1.0198084896893806026128701504490923825,1.0203841399637815859800148843988034350]	0.81394059530097648847023709411713210678	1.2157235720420877559020524307428422860	2.0296641673430642443722895248599743928
[1.0203841399637815859800148843988034350,1.0209838504047870123617712618755913944]	0.81144297773584477563820630115147896485	1.2190818157825700553976319369271859390	2.0305247935184148310358382380786649039
[1.0209838504047870123617712618755913944,1.0216089181712480650668543105377358770]	0.80888361356806837904127114795317718334	1.2225376918059943107353022393430678300	2.0314213053740626897765733872962450133
[1.0216089181712480650668543105377358770,1.0222607253243019778727319845508808739]	0.80626037319657372530545248150147304218	1.2260952581383868167764285083302177775	2.0323556313349605420818809898316908197
[1.0222607253243019778727319845508808739,1.0229407453110393816747252804485091080]	0.80357103671173526549272943869993897135	1.2297587887853995968202751140695415326	2.0333298254971348623130045527694805039
[1.0229407453110393816747252804485091080,1.0236505500092032812275168974637413757]	0.80081328961283975774357464128174876617	1.2335327875563223600561941852313687319	2.0343460771691621177997688265131174980
[1.0236505500092032812275168974637413757,1.0243918173866247530789236360952824932]	0.79798471832563073587561290392201562525	1.2374220029072539288467636142899973801	2.0354067212328846647223765182120130053
[1.0243918173866247530789236360952824932,1.0251663398346637120142321565191013416]	0.79508280551396269280493927985240068802	1.2414314438870032744200600978346356426	2.0365142494009659672249993776870363307
[1.0251663398346637120142321565191013416,1.0259760332410954793552597139587648448]	0.79210492518015057779236084561050833724	1.2455663972766549557557290625426658691	2.0376713224568055335480899081531742063
[1.0259760332410954793552597139587648448,1.0268229468747337797066489654312573984]	0.78904833754933707342595928803179624116	1.2498324460217662271219419610320465379	2.0388807835711033005479012490638427790
[1.0268229468747337797066489654312573984,1.0277092741616840471733538714187004844]	0.78591018373415214786050631830673873266	1.2542354890649209897008901151396717349	2.0401456727990731375613964334464104676
[1.0277092741616840471733538714187004844,1.0286373644415615706902234705470056658]	0.78268748017714664270301000807670565149	1.2587817626959104455230471142372449497	2.0414692428730570882260571223139506012
[1.0286373644415615706902234705470056658,1.0296097358013799925525756670624200281]	0.77937711286999081286762209546757157881	1.2634778635472066703103988467854881338	2.0428549764171974831780209422530597126
[1.0296097358013799925525756670624200281,1.0306290890952195886629780145889481739]	0.77597583135029401864347823045977945448	1.2683307733737115514334668676711140209	2.0443066047240055700769450981308934754
[1.0306290890952195886629780145889481739,1.0316983232693345894987286976768384997]	0.77248024247918613216759195325283311298	1.2733478857680706707661644934193873447	2.0458281282472568029337564466722204576
[1.0316983232693345894987286976768384997,1.0328205521251786340966650566292107509]	0.76888680400557760267936399620671028339	1.2785370349762129321596445622286357226	2.0474238389817905348390085584353460060
[1.0328205521251786340966650566292107509,1.0339991226670531146850902315054177338]	0.76519181792636790014487357396867489408	1.2839065269922863870471714536193379224	2.0490983449186542871920450275880128165
[1.0339991226670531146850902315054177338,1.0352376351968627648258005421665602503]	0.76139142365589863713890005870725601550	1.2894651731278829496975708958426491057	2.0508565967837815868364709545499051212
[1.0352376351968627648258005421665602503,1.0365399653359570808809833332014791715]	0.75748159102276029278897797508549915523	1.2952223262674516297325890774665380098	2.0527039172902119225215670525520371650
[1.0365399653359570808809833332014791715,1.0379102881734184499543320468920358792]	0.75345811311778916473296618047650712422	1.3011879200401591828032003846033489328	2.0546460331579483475361665650798560570
[1.0379102881734184499543320468920358792,1.0393531047616139854210619312271099928]	0.74931659902388193544773257931799241930	1.3073725111582286690605623557725801888	2.0566891101821106045082949350905726081
[1.0393531047616139854210619312271099928,1.0408732712035554138031607314011671721]	0.74505246646627830004762365822973481054	1.3137873251930185162285866189047572026	2.0588397916592968162762102771344920131
[1.0408732712035554138031607314011671721,1.0424760306028173264290187726763436190]	0.74066093443141042561604223584167487275	1.3204443060828284758492447261360135838	2.0611052405142389014652869619776884566
[1.0424760306028173264290187726763436190,1.0441670481756636877936418999437852282]	0.73613701581351083260840671573369091552	1.3273561696906417957819606153012462716	2.0634931855041526283903673310349371871
[1.0441670481756636877936418999437852282,1.0459524498568445164809057876894728148]	0.73147551016115569351280241539697066228	1.3345364617557103646938796176619426806	2.0660119719168660582066820330589133428
[1.0459524498568445164809057876894728148,1.0478388647654664852331291010668375878]	0.72667099661107796663016432467996982367	1.3419996206099942771891588074270116738	2.0686706172210722438193231321069814975
[1.0478388647654664852331291010668375878,1.0498334719356212974478810983300981185]	0.72171782711422726504220876257733900182	1.3497610450588563660431424681166214243	2.0714788721730836310853512306939604261
[1.0498334719356212974478810983300981185,1.0519440517582636003540569835910154469]	0.71661012007952936241505331123817649930	1.3578371678548912101774251231627441001	2.0744472879344205725924784344009205994
[1.0519440517582636003540569835910154469,1.0541790426263229998776733053193347721]	0.71134175458449270346920469309353383351	1.3662455352240510296494622453676704243	2.0775872898085437331186669384612042578
[1.0541790426263229998776733053193347721,1.0565476033243194812926713868485457507]	0.70590636532914357581416017329200666810	1.3750048929339164209990964292032084571	2.0809112582630599968132566024952151252
[1.0565476033243194812926713868485457507,1.0590596817568622221168895456222651010]	0.70029733854120174459045372398359051387	1.3841352794245023599443739844070956205	2.0844326179657041045348277083906861344
[1.0590596817568622221168895456222651010,1.0617260906672799363593568720243567478]	0.69450780907642096880290674995123752762	1.3936581265516643204805273142083101175	2.0881659356280852892834340641595476451
[1.0617260906672799363593568720243567478,1.0645585910580481305443912905137838314]	0.68853065899912377290703951455432353082	1.4035963685210299226974095139155808020	2.0921270275201536956044490284699043328
[1.0645585910580481305443912905137838314,1.0675699840882496131939715886390479885]	0.68235851797467853299761143638855516515	1.4139745596152125289010323893455557063	2.0963330775898910618986438257341108715
[1.0675699840882496131939715886390479885,1.0707742122893882889405277399232640618]	0.67598376585851474018772944935946513850	1.4248190013373217246046184259920210800	2.1008027671958364647923478753514862185
[1.0707742122893882889405277399232640618,1.0741864710085137806789930380752224675]	0.66939853792573346004641306350904856454	1.4361578796075352616278076927397574975	2.1055564175332687216742207562488060621
[1.0741864710085137806789930380752224675,1.0778233310554407708568393890398981795]	0.66259473325186265781657166936638829980	1.4480214126543326301332952773128415929	2.1106161459061952879498669466792298927
[1.0778233310554407708568393890398981795,1.0817028735969847993445873307945486160]	0.65556402682913586163276542259894742420	1.4604420102349538371791330565338685509	2.1160060370640896988118984791328159751
[1.0817028735969847993445873307945486160,1.0858448384030695296683213929354602798]	0.64829788608396579350987604398792519169	1.4734544447971352785207059079208605729	2.1217523308811010720305819519087857646
[1.0858448384030695296683213929354602798,1.0902707866039864006917875135733655109]	0.64078759254991169622440593176546589926	1.4870960351518429760461783115427007009	2.1278836277017546722705842433081666001
[1.0902707866039864006917875135733655109,1.0950042791607465023798607920418416169]	0.63302426954590568628497655322265400961	1.5014068431593870556906132616024137635	2.1344311127052927419755898148250677731
[1.0950042791607465023798607920418416169,1.1000710722759490208987626282503614164]	0.62499891681081812579176732411249809697	1.5164298838328428347569566659149892011	2.1414288006436609605487239900274872980
[1.1000710722759490208987626282503614164,1.1054993309741373554048168941824980272]	0.61670245315095135922272688374880111681	1.5322113491259969158248258256138675481	2.1489138022769482750475527093626686649
[1.1054993309741373554048168941824980272,1.1113198620498863229328571682910547740]	0.60812576826423912437901246420297099137	1.5488008454898997864716590619333065030	2.1569266137541389108506715261362774943
[1.1113198620498863229328571682910547740,1.1175663675087381436378456407328371523]	0.59925978501017730206126242830719935513	1.5662516450433078444455868518046255754	2.1655114300534851465068492801118249305
[1.1175663675087381436378456407328371523,1.1242757194984819426697048570605309962]	0.59009553349282142454494259114893435481	1.5846209498976386672858576740118903939	2.1747164833904600918308002651608247488
[1.1242757194984819426697048570605309962,1.1314882575319437835957780073971747926]	0.58062423840886305807929072795688849478	1.6039701687955754953368821486774080775	2.1845944072044385534161728766342965722
[1.1314882575319437835957780073971747926,1.1392481085210732683706174661643535992]	0.57083742117510906683324109123909337361	1.6243652047527607213264898429964258264	2.1952026259278697881597309342355192000
[1.1392481085210732683706174661643535992,1.1476035297573168096324217693703984575]	0.56072701837851431952506089611361619626	1.6458767518228974714084480788452438859	2.2066037702014117909335089749588600822
[1.1476035297573168096324217693703984575,1.1566072744650491923717959517961661233]	0.55028551807341835686879145232317855003	1.6685805984278380964650372361066624362	2.2188661165012564533338286884298409862
[1.1566072744650491923717959517961661233,1.1663169789022470964513391215252180257]	0.53950611536798067292661688642024879193	1.6925579338979199145801140723995442619	2.2320640492659005875067309588197930538
[1.1663169789022470964513391215252180257,1.1767955691649109693716069894749147763]	0.52838288857504308945403940447107301675	1.7178956539498391882568315216055668082	2.2462785425248822777108709260766398249
[1.1767955691649109693716069894749147763,1.1881116848502450634429737912655990699]	0.51691099692877367649882101561148995772	1.7446866597917019504760649490711237752	2.2615976567204756269748859646826137329
[1.1881116848502450634429737912655990699,1.2003401155340529257276796141453046887]	0.50508690046184274338177655016084721109	1.7730301443982717501724750359290257840	2.2781170448601144935542515860898729951
[1.2003401155340529257276796141453046887,1.2135622446138303233845631553480325355]	0.49290860207120037256565194513359682965	1.8030318582665829456936026090184559559	2.2959404603377833182592545541520527855
[1.2135622446138303233845631553480325355,1.2278664934664841430506363559242137332]	0.48037591104624206982736718047719898778	1.8348043456815107641903939444584356752	2.3151802567277528340177611249356346630
[1.2278664934664841430506363559242137332,1.2433487570918447393859515407464876824]	0.46749072636593639427588211328208405650	1.8684671412509307744204449478934193889	2.3359578676168671686963270611755034454
[1.2433487570918447393859515407464876824,1.2601128205072184117761227780817282577]	0.45425733687158106125350583506566486130	1.9041469152929729357020274351579588018	2.3584042521645539969555332702236236631
[1.2601128205072184117761227780817282577,1.2782707432015080826950912644965998229]	0.44068273397952301525908984464818636613	1.9419775556831063971149819004250092981	2.3826602896626294123740717450731956643
[1.2782707432015080826950912644965998229,1.2979431970640633221197772750747269097]	0.42677693091931694639393231194788019261	1.9821001731352990570664809077622339871	2.4088771040546160034604132197101141797
[1.2979431970640633221197772750747269097,1.3192597415296350868410926255402607507]	0.41255328059544996687206281356447785092	2.0246630167672482137511735953919082727	2.4372162973626981806232364089563861236
[1.3192597415296350868410926255402607507,1.3423590184275970125727698293479425769]	0.39802878213215432845556997102290638114	2.0698212873776684301559487490292630488	2.4678500695098227586115187200521694299
[1.3423590184275970125727698293479425769,1.3673888484349597349968443330501487534]	0.38322436406412060498350917451096795302	2.1177368373527874598249768272031175714	2.5009612014169080648084860017140855245
[1.3673888484349597349968443330501487534,1.3945062113874206353222581396540901750]	0.36816513011545446903120660016191937177	2.1685777487290129130334585108450667519	2.5367428788444673820646651110069861237
[1.3945062113874206353222581396540901750,1.4238770942975087747083227133322020871]	0.35288055174267664852153095439670696469	2.2225177848566198218933554683691414330	2.5753983365992964704148864227658483977
[1.4238770942975087747083227133322020871,1.4556761940516545369517885980437403998]	0.33740459032308286366617861820489647816	2.2797357164709432839676546872175910204	2.6171403067940261476338333054224874986
[1.4556761940516545369517885980437403998,1.4900864666503189448070843659717369924]	0.32177573129598635557747292912950964812	2.3404145298314128630755905788062316004	2.6621902611273992186530635079357412485
[1.4900864666503189448070843659717369924,1.5272985216660854124419408735705344167]	0.30603691297168046022507954273659293843	2.4047405328581845414120223978339848846	2.7107774458298650016371019405705778231
[1.5272985216660854124419408735705344167,1.5675098693287377642717478707985917142]	0.29023533435539534559130773860564136483	2.4729023846442612164516888626561245981	2.7631377189996565620429966012617659629
[1.5675098693287377642717478707985917142,1.6109240381152757987450568095143693246]	0.27442212938183284988470954051994582151	2.5450900839258360354102845827382397776	2.8195122133076688852949941232581855991
[1.6109240381152757987450568095143693246,1.6577495925068218033944594011627240537]	0.25865189951509975732664172791425068629	2.6214939624415705204342049559427207894	2.8801458619566702777608466838569714757
[1.6577495925068218033944594011627240537,1.7081990930077299486802615521816794814]	0.24298210269598672494342306773661533768	2.7023037388177598552906247342605018274	2.9452858415137465802340478019971171651
[1.7081990930077299486802615521816794814,1.7624880527121803550274714986817644640]	0.22747230389804253305847828497237035119	2.7877076967773281171376002254524287785	3.0151800006753706501960785104247991297
[1.7624880527121803550274714986817644640,1.8208339555905209405274270523770800957]	0.21218330068067287182013544789297011747	2.8778920571501502755973233091015991852	3.0900753578308231474174587569945693027
[1.8208339555905209405274270523770800957,1.8834554101274944343594086990932108253]	0.19717614551581720254684735359611523798	2.9730406155048459152193080223338229828	3.1702167610206631177661553759299382208
[1.8834554101274944343594086990932108253,1.9505715169251174331777185359228927416]	0.18251109458908446759935373584983224733	3.0733347155804295095802477659307162368	3.2558458101695139771796015017805484841
[1.9505715169251174331777185359228927416,2.0224015295571935097876197469495226549]	0.16824651944429334221953719535269557599	3.1789536227513780501771251349786236862	3.3472001421956713923966623303313192622

mathe · 发表于 2019-9-8 07:44:29

9#可以如下处理:
由于函数$\sqrt(m)-\ln({e*m}/{e-1})$导函数为$1/{2\sqrt{m}}-1/m$,所以唯一极值点为$m=4$,可以得出
$\sqrt(m)-\ln({e*m}/{e-1})>=2-\ln({4e}/{e-1})=0.15503049349302749014389211201631716198$
由此可以知道$0.15503049349302749014389211201631716198+m+\ln({e*m}/{e-1})\le m+\sqrt{m}$恒成立
也就是已经严格证明在$m\gt 2.0191639426920814670741038938393492242$时不等式成立

mathe · 发表于 2019-9-8 07:58:10

另外对于11#中，我们可以先证明$\ln(1+x)\lt x-x^2/2+x^3/3$
所以我们有$m=1+r_2-\ln(1+r_2)\gt 1+{r_2^2}/2-{r_2^3}/3$
另外可以证明在$0\lt r_2\lt 2.1407350339519857241589616899606612617$ 即方程 $9*x^3 + 96*x^2 + 112*x - 768$的根时，有
$\sqrt{1+{r_2^2}/2-{r_2^3}/3} \gt 1 + 1/4 r_2^2 - 1/6 r_2^3 - 1/32 r_2^4$
所以得出在$0\lt r_2\lt 2.1407350339519857241589616899606612617$时，$m+\sqrt{m} \gt 2 + 3/4*r_2^2 - 1/2*r_2^3 - 1/32*r_2^4$
另外一方面，我们可以比较函数$\ln({1+x}/{1-x})-2x$和${2x^3}/3+{4x^5}/5$.显然这两个函数在x=0都等于0
而两个函数差的导函数为$1/{1+x}+1/{1-x}-2-2x^2-4x^4={x^4(2x^2-1)}/{1-x^2}$,所以在$0<x<{\sqrt{2}}/2$时必然导函数之差小于0，所以必然有$\ln({1+x}/{1-x})-2x\le {2x^3}/3+{4x^5}/5$，所以${\ln({1+x}/{1-x})}/x-2\le {2x^2}/3+{4x^4}/5$
另外可以得出$0\lt x\lt 0.1359428875112682157 $即方程$399*x^2 + 240*x - 40$的根时，${2x^2}/3+{4x^4}/5\lt x^2/2 -x^3/3+ 1/4 x^2 - 1/6 x^3 - 1/32 x^4$,
由此得出$r_2$在这个范围时题目中不等式必然成立，对应的我们可以严格证明$1\lt m \lt 1.0084798435542516730003035811003272610$时不等式成立。
上面如果适当调整函数应该还可以找到更好的结果，但是最难的部分已经解决了

wayne · 发表于 2019-9-8 08:47:27

猜想: 引入调和函数$H_m$,其定义参考http://mathworld.wolfram.com/HarmonicNumber.html
$H_m = \gamma+\psi_0(m+1) . m\in \RR. \gamma= 0.57721566490153286060651209008, \psi_0(z) = \int_0^{\infty } (\frac{e^{-t}}{t}-\frac{e^{-z t}}{1-e^{-t}}) \, dt$ , 当$m$是整数的时候,$H_m = \sum_{k=1}^m\frac{1}{k}$

已知方程 `x-\ln x=m\;(m\in \RR)` 有两不等实根 $x_1,x_2$，那么 $x_1+x_2\leq m+H_m$, $H_m$是调和函数.

=============
可以发现,当$1<m<100$的时候,都是成立的,而且误差很小, 基本上可以确定$m+H_m$ 符合$x_1+x_2$的渐近规律 ,但是$m+\sqrt{m}$就飘的比较厉害了.
设$s=6.01008178667859777782319341185$, 其中 $1<m<s$的时候$1+H_m > \sqrt{m} $ , 而, $ m>s, H_m < \sqrt{m}$

Table[{N[m],ans=Quiet[FindRoot[x-Log[x]==m,{x,{1/m,m}},WorkingPrecision->30]][[1,2]];ans//Chop,{N[m+Sqrt[m]],N[m+Sqrt[m]]-Total[Chop[ans]]},{N[m+HarmonicNumber[m]],N[m+HarmonicNumber[m]]-Total[Chop[ans]]}},{m,1,100,1}]//Column

复制代码

以下列出来的分别是m值,两个根的值, $m+\sqrt{m}$及其误差, $m+H_{m}$以及误差

{1.,{1.00000000000000000000000000000,1.00000000000000000000000000000},{2.,0.},{2.,0.}}
{2.,{0.158594339563039362153395341988,3.14619322062058258523706102852},{3.41421,0.109426},{3.5,0.195212}}
{3.,{0.0524690974577148724098736112860,4.50524149579288336699862443214},{4.73205,0.17434},{4.83333,0.275623}}
{4.,{0.0186606290886833424583352788495,5.74903138601270155099097984403},{6.,0.232308},{6.08333,0.315641}}
{5.,{0.00678381135209697125412617571662,6.93684740722021872216431824129},{7.23607,0.292437},{7.28333,0.339702}}
{6.,{0.00248491933514945299249607853514,8.09071740515548459615806517541},{8.44949,0.356287},{8.45,0.356798}}
{7.,{0.000912714633504814127917301061536,9.22154230138681009493256216255},{9.64575,0.423296},{9.59286,0.370402}}
{8.,{0.000335575219738042512501307128058,10.3355936303148423319991846184},{10.8284,0.492498},{10.7179,0.381928}}
{9.,{0.000123425036886336175483768024472,11.4368396975756112254380665412},{12.,0.563037},{11.829,0.392005}}
{10.,{0.0000454019910564829645753546702237,12.5279632019821742536902940417},{13.1623,0.634269},{12.929,0.40096}}
{11.,{0.0000167019797440434829263998851584,13.6108686381498759384044381340},{14.3166,0.705739},{14.0199,0.408992}}
{12.,{6.14425010502158477465211174917*10^-6,14.6869600271174058810499258462},{15.4641,0.777135},{15.1032,0.416245}}
{13.,{2.26033451608740479504947909378*10^-6,15.7573040040643653551653533174},{16.6056,0.848245},{16.1801,0.422827}}
{14.,{8.31529410544441007698836956412*10^-7,16.8227310080755478255166167768},{17.7417,0.918926},{17.2516,0.42883}}
{15.,{3.05902414078098414577454207809*10^-7,17.8839009181026785098798459694},{18.873,0.989082},{18.3182,0.434328}}
{16.,{1.12535187383426801539188724114*10^-7,18.9413472159544750825669544634},{20.,1.05865},{19.3807,0.439382}}
{17.,{4.13993789017602045713198330598*10^-8,19.9955076298141846359677459668},{21.1231,1.1276},{20.4396,0.444045}}
{18.,{1.52299799766649167594461909916*10^-8,21.0467459620529743314604324511},{22.2426,1.19589},{21.4951,0.448362}}
{19.,{5.60279646892859563242738704595*10^-9,22.0953679935017959975536676865},{23.3589,1.26353},{22.5477,0.452372}}
{20.,{2.06115362668691209637687553309*10^-9,23.1416333028010367067760849799},{24.4721,1.3305},{23.5977,0.456106}}
{21.,{7.58256043366142214706130204112*10^-10,24.1857642040408054818753088686},{25.5826,1.39681},{24.6454,0.459594}}
{22.,{2.78946809364703803217682267177*10^-10,25.2279526103449451144338495321},{26.6904,1.46246},{25.6908,0.462861}}
{23.,{1.02618796327549520007192663362*10^-10,26.2683653778501610784960993621},{27.7958,1.52747},{26.7343,0.465926}}
{24.,{0,27.3071485184050865096259974250},{28.899,1.59183},{27.776,0.46881}}
{25.,{0,28.3444305578898612489009000639},{30.,1.65557},{28.816,0.471528}}
{26.,{0,29.3803252408283779388406484395},{31.099,1.71869},{29.8544,0.474094}}
{27.,{0,30.4149337288741634682380883803},{32.1962,1.78122},{30.8915,0.476523}}
{28.,{0,31.4483464031685521958216883566},{33.2915,1.84316},{31.9272,0.478825}}
{29.,{0,32.4806443535686381497178605099},{34.3852,1.90452},{32.9617,0.481009}}
{30.,{0,33.5119006180780928715177040139},{35.4772,1.96532},{33.995,0.483087}}
{31.,{0,34.5421812213127104690679389999},{36.5678,2.02558},{35.0272,0.485064}}
{32.,{0,35.5715460500147984914500639544},{37.6569,2.08531},{36.0585,0.486949}}
{33.,{0,36.6000495954739980370641547852},{38.7446,2.14451},{37.0888,0.488749}}
{34.,{0,37.6277415865008364483248857877},{39.831,2.20321},{38.1182,0.490468}}
{35.,{0,38.6546675318254898748650405654},{40.9161,2.26141},{39.1468,0.492114}}
{36.,{0,39.6808691870934320136975965980},{42.,2.31913},{40.1746,0.49369}}
{37.,{0,40.7063849587374473209499951865},{43.0828,2.37638},{41.2016,0.495201}}
{38.,{0,41.7312502547280539759680007470},{44.1644,2.43316},{42.2279,0.496652}}
{39.,{0,42.7553497575690141632778713466},{45.245,2.48965},{43.2535,0.498193}}
{40.,{0,43.7791578586257387880687275326},{46.3246,2.5454},{44.2785,0.499385}}
{41.,{0,44.5995795038245362930915736489},{47.4031,2.80354},{45.3029,0.703354}}
{42.,{0,45.4967438001310664543099636824},{48.4807,2.984},{46.3267,0.829999}}
{43.,{0,46.7961162955251146948984774252},{49.5574,2.76132},{47.35,0.553882}}
{44.,{0,47.8451699089225101414645554405},{50.6332,2.78808},{48.3727,0.527556}}
{45.,{0,48.8600924373465432530362428681},{51.7082,2.84811},{49.3949,0.534856}}
{46.,{0,49.8613148461776324842926875542},{52.7823,2.92102},{50.4167,0.555372}}
{47.,{0,50.7056297741095076452565598902},{53.8557,3.15002},{51.438,0.732334}}
{48.,{0,51.3391788318981184359386733443},{54.9282,3.58902},{52.4588,1.11962}}
{49.,{0,52.9098775041914210754064666916},{56.,3.09012},{53.4792,0.569328}}
{50.,{0,53.1845251279361158497586600963},{57.0711,3.88654},{54.4992,1.31468}}
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复制代码

mathe · 发表于 2019-9-8 08:52:45

你这个在m不是整数时如何定义？

mathe · 发表于 2019-9-8 10:16:40

H_m的上述定义在m=1处导数略大于sqrt(m),所以在m=1处估计的没有sqrt(m)好。sqrt(m)在m=1的导数为1/2. 合法的最佳估计必然导数不小于1/3

mathe · 发表于 2019-9-8 12:50:44

wayne的结论应该也不难验证，事实上对于任意c>0,对于充分大的m，两根和都必然小于m+ln(m)+c

kastin · 发表于 2019-9-8 14:25:19

换个思路，原方程可写为 `\int_1^{x_2}(1-\frac 1t)\dif t=x_2-\ln x_2=m , \int_1^{x_1}(1-\frac 1t)\dif t=x_1-\ln x_1=m.` 这样就能通过积分的方向单独估计 `x_1,\,x_2` 了。积分方面有一些数值公式和余项估计，或许是一个思考方向。

mathe · 发表于 2019-9-8 16:05:59

对于wayne的结果，我们先找出关于$psi(x)$的不等式，有$\psi(x+1)\gt \ln(x)$
而在9#中我们有不等式$x_2\lt m+\ln(m)+0.45867514538708189102164364506732970188$
于是对于$x_1\lt 0.11854051951445096958486844501507272917$ wayne的不等式已经必然成立
而在另外一个端点，注意到m接近1时,digamma的导数大于sqrt{m}的导数，而且digamma的导函数和sqrt{m}的导函数都是严格递减的，所以我们可以digamma导数取值大于1/2的这段区间，这段区间内digamma{m}+Euler必然比sqrt{m}增加的快，从而严格大于它。于是可以找到在m=1附近一段区间的wayne不等式成立。
同样，中间区域可以类似上面方法数值计算验证

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