JiHreir maRna daKnu cera v JaroSmiimrs a Equatiaonnds Inequalities ElemenPtraorbyl aenmds TheorienmA sl geabnrda Number Theory TranslabtyeK da rDli lcher �'z._� Canad1Maant hematSioccaile ty � :7 Sociemtaet hematidquuC ea nada CMBSo oikMnsa thematics Ouvramgaetsh edmea dteIiS aqM uCe s EquatiaonndsI nequalities 1 HERMAN /KutERA/SIMSA AbeliGanr oupansd RepresentoaftiF oinnsiP taerti allOyr derSeedt s 2 ARNOLD ConveAxn alyasnidsN onlinOepatri mization 3 BORWEIN/LEWIS OrthogoPnoallyn omiaflosExr p onentiWaeli ghts 4 LEVIN/LUBINSKY ReflectGiroonu pasn dI nvariTahneto ry 5 KANE JifiH ermanR adaKnu cera v JaromSiirm sa Equatainodn s Inequalities ElententParroyb leantnsd Theoreinnt sAl gebraan d NUinberT heory TranslbaytK eadrD lil cher Springer Herman RadanK ucera Jiff Brno Gr ar School DepartmoefnM ta thematics amm KptJ.ar os1e4 FaculotfyS cienMcaes,ar ykU niversity tf. 658 Brno Janackonvaom .2 a 70 CzecRhe public 6629 5B rno CzecRhe public Jaromfr imsa S Translator MathematiIcnaslti tute Karl Dilcher Academoyf S ciencoefths e Czech DepartmoefnM ta thematiancds S tatistics Republic DalhouUsniiev ersity iZk.ova2 2 HalifNaoxv,a S cotiaB 3H 3J5 Z 6166 2B rno Canada CzecRhe public Editors-in-Chief Redacteurs-en-chef JonathaBno rwein PeteBro rwein Centref orE xperimenatanld C onstructMiavteh ematics DepartmoefnM ta thematicasn dS tatistics SimoFnr asUenri versity Burnaby,B ritisCho lumbiVaS A 1S6 Canada MathematSiucbsj eCclta ssific(a2t0i0on01 )6:G 202,0Kx x Libraoryf C ongresCsa taloging-in-PDuabtlai cation Herman, Jiff. [Metodyr esemnaft ematiculcylcohIh . E nglish] Equationasn di nequal:ie tlieemse ntparroyb leamnsd t heoreimnsa lgebarnadn umber theoryJ iffH erman,R adaKnu eerJaar,o mirS ims;a translabtyeK dar lD ilcher. I p.e m.- (CMSb ookisn m athematics) Includbesi bliograrpehfiecraelna cnedis n dex. ISBN0 -387-98942-p0a per) (alk. 1.E quations-Numeriscoallu tio2n.sIn .e qualiti3e.Prs o.b lems olvinIg.K. u cera Radan. SimSaJ,ar omir. Title. Series. n. m. IV. QA218.H471230 00 512.4-9-d'c21 99-047384 ona cid-frpeaep er. Printed 2000 SpringeVre-rlaNge wY orkIn,c . © Allr ighrtess erveTdh.i sw orkm ayn otb etr anslaotrec do piedw holoer part withotuhte in in writtpeenrmi ssioonft he publ(iSsphrienrg Veerr-laNge wY orkIn,c .1,7 5F iftAhv enueN,e wY ork, 10010U,S A),e xcepfto rb rieefx cerpts connectwiiotnrh e vieowrss cholaralnya lysUisse. NY in inc onnectiwoint ha nyf orm ofi nformatsitoon raganed retrievealle,c troanidca ptatcioomnp,u ter softwareo,r b ys imiloarrd issimilmare thodolnoogwykn own orh ereafter developiesfd o rbidden. The useo fg enerdaels cripntaimveestr ,a den ames,t rademarektsc,. ,t hipsu blicaetvieoinn, ft he in formerar e note speciiadlelnyti fieidsn, o tt ob et akeans a sigtnh astu cnh amesa,s u nderstboyo d theTr adMear ksa ndM erchandMiarskes A ctm,a ya ccordibnegu lsye frde elbyy a nyone. Productimoann agebdy A llaAnb ramsm;a nufactusruipnegrv isbeydJ eromBea sma. Photocompocsoepdyp reparefrdo m thetr anslator'sfi les. UT:EX Printaenddb ounbdy R .RD.o nnelleayn dS onsH,a rrisonbuVrAg., Printientd h eU niteSdt atoefsA merica. 9 8 7 6 5 4 3 2 1 ISBN0 -387-989S4p2ri-n0g er-VeNrelwa Ygo rkB erliHne idelberSgP IN 10745911 Preface Thisb ooki si ntendase da t ext fora problem-soclovuirnsgae tt hefi rst­ ors econyde-aurn iverslietvyeas l ,a t extf ore nrichmcelnasts esf otral ented high-soclhs otudenotrsf ,o rm athematciocsm pteititornai ninIgtc. an also be as as ourcoefs upplementmaartye rifalo ran y coursdee aling with used alegbraeiqcu atioorni sn equaliotrit eoss u,p plemean stt andaerlde mentary numbetrh eoryc ourse. Theraree alreamdanyy excellbeonotk osn t hem arketth actan beu sed foar problem-soclovuirnsgHe o.w evesro,m ea rem erelcyo llectoifpo rnosb ­ lemsfr om a varieotfyfi elds alacnkdc ohesiOotnh.e rpsr esepnrto blems accorditnotg o pibcu,t p rovildiet tolren ot heoretbiaccalk rgoundM.o st problebmo oks haal viem itneudm beorf r athcehral lengipnrgo blemWhsil.e these probletmesn dt ob eq uitbee autifutlh,e cyan appeafro rbiddanidn g discoguirnatgo a beginnisntgu denetv,e nw itwehl l-motivaatnedcd a refully writtseonl untsio. a conseqnucees,t udenmtasy decidteh atp roblem As solviinsg onfloyrt hef ew hipgerhf ormeirnts h eir calnasdsa ,b andotnh is impotrantp arto ft heimra thematicaaln,d i needd overaleld,u cation. Oneo ft her easonwsh yp roblesmo lviinsog ft enf ountdo b ed ifficulits thef actth ait nr ecendte cadest here beenl esse mphasiosn t echnical has skillisnN ortAhm ericanh igh-schmoaotlh ematiFucrst.h ermorseu,c shk ills arer areltya ughatt u nivresitwyh,e rmeo stc oursaerseq uitteh eoretoirc al structure-oriean rteseudl.ta , l acokf ' 'mathematfliucealn cyi"so ften As evideenvte inn u ppery earastu nivseirttyh;i rse ductesh ee njoymeonftt h e subjeacntd i mpaiprrso gresansd success. A seconrde asoni st hamto sts tudtesna ren otu sedt om orec omplexo r multiy-elraepdr oblemsI.tw ouldc ertainbleyw rongt og ivteh ei mpres- Preacfe vi siotnh aat lmla thematipcralo blesmhso ulsdu ccumtbo a straightforward approacIhn.d eemdu,c ho ft hea ttractivoefnm eastsh ematilcisesi nt he satisfacdteiroinv ferdo ms olvidnigffi culptr obleaftmesr muche fforatn d severfualt ilaet temptOsn. t heo thehra ndb,e in"gs utck"t ooo ften and fort ool oncg an bvee ryd isocuragianngd c ounterproduancdt iivsue l,t i­ matelay w astoef t imet hact oulbde b ettesrp enlte arning panrdac ticing new techniques. Thisb ooka ttempttosa ddretshses ei ssuaensd o fferas p artirale med.y Thisi sd oneb y emphasizinbgas ica lgebroaipce ratianodn so thetre chni­ caslki lltsh aarte reinforicnne udm erouesx amplaesn dex ercisHeosw.e ver, evetnh ee asiesto nesr equiar sem altlwis t.W e thereforet hhaottph ei psr o­ cesosf p racticidnoge sn otb ecompeu relryo teb,u tw ilrle tatihne r eader's interesrta,i shei osr h erl evoeflc onfidenacned,e ncoureaa gttempttoss olve someo ft hem orec hallenpgrionbgl ems. Anotheari mo ft hibso oki st of amiliartizhee r eadewri thm ethodfso r solvinpgr obleimnse lementmaartyh ematiaccsc,ei sbslteo b eginningu ni­ versiantdy advancedh igh-schosotlu denTthsi.s c an bed onei nd ifferent ways;f ori nstantchee,a uthorosfs omeb ooksi ntrodusceev ergale neral methods( e.,gi .nductianoanl,og y,o rt hep igoenholper incipanlde )i llus­ trateea cho new ithc oncreptreo blefrmosm differeanrte aso fm athematics andw ithva ryindgeg rees ofd ifficuyl.t Oura pproachho,w eveirsd, i ffereWnet p.r esenat r elatisveellfy- condt aine overviewo vers omep artosf e lementamrayth ematitchsa td on ot receive mucha ttentiionhn igh-schoanodl universietdyu catiWoen .g iveo nly enought heoretibcaalc gkrountdo m aket hesteo piscesl f-contanainde dr ig­ orousan,d concnetratoen s olvipnagr ticuplraorb leimnst het hreem ain areasc orrespondtiont gh efi rstth reec hapteorfts h ibso ok. Thesec haptearres f airly indepoefne daecnhot t hewri,t ho nlay limited numbeor fc ross-refeWreitnhciesen.a ch chatperc,l ustoerfss e ctioannsd subscetioanrse t ietdo getheeri thbeyrt opiocr,b yt he methnoeeddesd to solvteh ee xamplanesd exerciTsheesp .r oblem-bcohoarka ctoefrt hist ext isu nderlinbeydt hel argen umbeorf e xercitsheescy;an bes olvebdy using a methoodr m ethodpsr eviouisnltyr uocdedW.e sugesgt thatt her eader firsctar efullsyt udayn yr elevanexta mplesb,e foraet tempttionsg o lvaen y 330 oft heex ercises. 760 Thei ndividpuralo ble(mdsi vidiendta op prxoimately examplaensd exercisaesr) e voafr yidnegg reeso fd iffictuylfr, om completsetlrya ight­ forwarwdh,e rteh eu seo fa methoudn decro nsirdaetiowni lilm mediately leatdo a solutitoonm ,u chm ored ifficulptr oblewmhso seso lutiownisl l sometimreesq uicroens ideraebffloer tT.h em ored emandinegx erciasrees (.* ) markedw itha na steriskA nswertso exercsiesc an bef ounidn t he all finalch atperw,h eraed ditiohnianltan sd instructtioto hnesm ored iffictu l oneasr eg iveasn well. Thep roblemwesr et akenfr om a varieotfym ainly EastEeurrno pean sources, as sMuacthh ematiOclaly mpiaadnsd o ther competMiatniyo ns. Preacfe vii otfha terhmene oortteeh fasaeesoicrwsrtlcitieoyEshb neelegl eai ksihn-gs p re.Aai ndm ecprrofirotstreean orltptfie rh ocoetntb th ilhaoeetnmi sr soslhu,iot ncui iblopapednrcil sitcbeh noesl,i se ghs -t.sWu ced heonotls was bletiehewvavtietelhtn iih oms cni sanetu acstcticesiorsmnefuu alltlaiytv ee woimrnak t anhidel mildatuitssanvi rtdercir .ssca i httneye ss Oonofeou bvrhjes base tee csontta eirlrestanspa pttsroioo vlaevc ihnesg ag ipvrSeoonbm wlpeteeri msom.edve viitsesffd roeeialor nlune teni ons plaaocttet h,iwre meerwet t hspoui r irlaolnne bap ltaoteerfhmrt e s book. Tbhoiiatoss rak ianto otsfsnhl e eCc zeoedoncifdht ion I" Mfesotorhmtl oahvdMeipestorimrd ryesIgao ne-t" tbilceamls pmaitueimsicba1k'gthcnulha9l Moe ht9 asd 6Ua nriiyBnvkrA e npfroarrsc.toio mtr yrt ehcet ion ( ) osfom mimenit otssrhp,m er a ihtmbnaseailee runenin ft acT lhh arneeg ed. shsoewrc"ittct iohoym npppsreea o"ttteb hi onlettfdce hip shmeot as ne -rts wdeterbelesude oot,tmf he es weie nrpcerio otnrmbhtpaleooie rnma st ed bootdfbhy oe ao lkp.ih nwaadasbde edxte idc.a l TChzeee docitfhth iiosun bsbstyeo ah douei kts n hp hoe­arvse been An ceinanrclasteil sf scso hermc eosntadunarydMd-ea sntfcthohsero m oalt ical OltyrmitaipnCnhi zReieaepn dIcug hthab asb ll uesioseocarnen . le d arg u basaits UM nasiicavnore ufuyrurtkmsrs aieett tshye ea afmt.Coa hrte ircss sescco.hpn ordAoeaE llrnvsiygem oltrififihsn twsreiacohs orh tnya pters wsasu ctcroebiutsyetths sd reful ia alannn lte cyowr prler yao tbelde m­ socloivtuniFhrl noaesl1gf e 9 D 9a8Ul vnhaeioitrHun sals iiitfeyax ,, CaInwtfaas ted thlca ato.ap u abrraosttsneblhe owyidoo s qkrt uke eid weelivlacne wl naias w tsrih ado menfa g tebh aencmadkansgtd ri ocual ab.Ti bhl'osesio tutktrrihueeseexc, at wanmtodprhr laieklesnn ee,g dv e e l odfil offittfecyh xu eme akeirp tca irwstseesiulas cisl utoa fleu oadrrr c ley asisgrneaneaddd o iehrncxokgiem ssre esw. Tthreaa flnlteesssthflol ,oaa l ttoCheowzexecria h ntm gbhp oilosek , myba seu fiaoct roiaa utnbc re lhesaeede rpu srcf'oraho gt irigamoh n­ scmhaottoahelc.eF h miweanhertato stitlhpr chlaeeswey afi i ,tdnl hedlir s boaro sikocou uhfsir tedc,ifueee tlqn ni ausiesna,ndt ie ieqosu.a lit Brannod Halifax JHiefrim an J1u9l9y9 RaKduacne ra Ja rSoimmi8ra KDairllc her Contents Preface v Symbols xi Alg1eb FrraiIocd mefntuPanoiwdtE oliqre.euass.a r t i so.ns . . 1. 1 1 2 FiSn.i .t e. .. . . . 5. 3 Poal.ly. sn. . .o . . m. i .. . . 23 ums 4 SmyemPtoorlmiynica. .l s.. . . . . . . . . . 5 SyosEfttq eiu.moa .ns. s . . .4368 . . . . . . . . . . . . . . 6 IrErtqaiu.to.a in. .os . n . a l. 6.0 . . 7 SoAmptepio Clofoins Nmcm upableerxs 7 5 . . . . . . . . Al1ge DbfirInaieecnq uaaliitPniteersdi ro otp.n.i s es . . ..9 0.. .. . 2 89 2 BiasMcte h..o . d. .s. . . . . . . 9.4 3 TUhseAe l Fgrooeasmfb u. r.l ai ..c ..1 0..6 . . 4 TMhhetedooS fq u.ar. e .s.. . .. . 1 3.1 . . . . . . . 5 TDhrieais anmcCntiuayd ncI' hnste. yq. u a1l2i4 . . . 6 TIhnePd ru.ic nt.ci. .io .pn. l e15 3 . . . . . 7 Ch'esIeb nqy.us ah.lev.. i ...t y. 15 4 . . IenqBueeetwaMnles ia t.ni es. .15 0. . . . 89 ApopIner nNrmduabi.tex ir osna1l6 9 . Contents x NumbeBrT chasCeo iory. n c.e.. p .t s.. . 3 117473 1 PrNium.m.e b . e.. r .s. 2 Cocn.eg s.r. . u. . .e . . n .. 138 . . . . . . 3 ConiOgrnVnu iareea.. n b .c. .l ..e. e s 189 . . . . . . . . 4 DoipEhqaun.at ti.in o..e. n s2 10 . . . . 5 SoloDvafib oEip.aql ht.ui.. a it noytn is2n 17e . . . 6 InPtareFrtga ePcartan ird. ot .. n ..a l23. 8 . . . 7 BReasper esen.t .a tio25n1 s . 8 DritiP'crhs.il n.e..c. . i. ..p le 258 . . . . 9 Pol.y .n o.m ials 268 . 10 . . . 274 . . . . HintsanH dAn iaswAnenrsntd tssC ow h earpst er 4 283 1 HiAnnttssCoaw h p1ae. trn. e.sd.r 238 2 HiaAnnntdswsC ae hp2r tse rt o 294 3 3 . 312 . Bibliography 337 Index 341