ShigeruFuruichi,HamidRezaMoradi AdvancesinMathematicalInequalities Also of Interest NonlinearEvolutionEquation Guo,Boling/Chen,Fei/Shao,Jing/Luo,Ting,tobepublished2020 ISBN978-3-11-062467-0,e-ISBN(PDF)978-3-11-061478-7, e-ISBN(EPUB)978-3-11-061547-0 Space-TimeMethods ApplicationstoPartialDifferentialEquations Ed.byLanger,Ulrich/Steinbach,Olaf,2019 ISBN978-3-11-054787-0,e-ISBN(PDF)978-3-11-054848-8, e-ISBN(EPUB)978-3-11-054799-3 DifferentialEquations AfirstcourseonODEandabriefintroductiontoPDE Ahmad,Shair/Ambrosetti,Antonio,2019 ISBN978-3-11-065003-7,e-ISBN(PDF)978-3-11-065286-4, e-ISBN(EPUB)978-3-11-065008-2 GameTheoryandPartialDifferentialEquations Blanc,Pablo/Rossi,JulioDaniel,2019 ISBN978-3-11-061925-6,e-ISBN(PDF)978-3-11-062179-2, e-ISBN(EPUB)978-3-11-061932-4 Shigeru Furuichi, Hamid Reza Moradi Advances in Mathematical Inequalities | MathematicsSubjectClassification2010 Primary:47A63,47A60,46L05,47A30,47A12;Secondary:26D15,15A60,94A17,15A39,15A45, 26D07,47B15,47A64 Authors Prof.ShigeruFuruichi Dr.HamidRezaMoradi 3-25-40Sakurajousui DepartmentofMathematics 156-8550Tokyo PayameNoorUniversity Japan 19395-4697P.O.Box [email protected] Tehran Iran [email protected] ISBN978-3-11-064343-5 e-ISBN(PDF)978-3-11-064347-3 e-ISBN(EPUB)978-3-11-064364-0 LibraryofCongressControlNumber:2019955079 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2020WalterdeGruyterGmbH,Berlin/Boston Coverimage:porcorex/E+/gettyimage.de Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck www.degruyter.com Preface Inmathematics,thewordinequalitymeansadisparitybetweentwoquantities,which isusedtoreflectthecorrelationbetweentwoobjects.Aninequalityimpliesthattwo amountsarenotequal.Inthe19thcenturyandwiththeemergenceofcalculus,the touchoftheinequalitiesanditsroleincreasinglybecameessential. Inmodernmathematics,inequalitiesplaysignificantrolesinalmostallfieldsof mathematics.Severalapplicationsofinequalitiesarefoundinlinearoperators,partial differentialequations,nonlinearanalysis,approximationtheory,optimizationtheory, numericalanalysis,probabilitytheory,statistics,andotherfields. Thebookmaybeusedbyresearchersindifferentbranchesofmathematicaland functionalanalysis,wherethetheoryofHilbertspacesisofrelevance.Sinceitisself- containedandalltheresultsareentirelyproved,theworkmayalsobeusedbygrad- uatestudentsinterestedintheoryofinequalitiesanditsapplications. Forthesakeofcompleteness,alltheresultspresentedarewhollyproved,andthe originalreferenceswheretheyhavebeenfirstobtainedarementioned. Finally,wewouldliketothanktoDr.Y.Tsurumi,andDr.F.-C.Mitroi-Symeonidis forcarefulcheckingtoimproveourmanuscript.Inaddition,thisbookwillnotbepub- lishedwithoutcooperationofallcoauthorsinourpapers.Sowewouldliketothank toallpreviouscoauthors.Wealsowouldliketothanktoourfamilysincetheygaveus timetowritethisbookwithasmile. https://doi.org/10.1515/9783110643473-201 Contents Preface|V 1 Introductionandpreliminaries|1 1.1 Introduction|1 1.2 Preliminaries|2 2 RefinementsandreversesforYounginequality|9 2.1 ReversesforKittaneh–Manasrahinequality|12 2.2 YounginequalitieswithSpechtratio|14 2.3 ReverseYounginequalities|17 2.4 GeneralizedreverseYounginequalities|19 2.5 Younginequalitiesbyexponentialfunctions|23 2.5.1 ElementaryproofsforDragomir’sinequalities|23 2.5.2 Furtherrefinementsbyr-exponentialfunction|28 2.6 Younginequalitieswithnewconstants|32 2.7 Moreaccurateinequalities|36 2.7.1 Inequalitiesforpositivelinearmaps|41 2.7.2 Inequalitiestooperatormonotonefunctions|42 2.8 Sharpinequalitiesforoperatormeans|46 2.8.1 Operatormeansinequalities|47 2.8.2 Younginequalitiesbysharpconstants|53 2.9 ComplementtorefinedYounginequality|57 2.9.1 RefinementofKantorovichinequality|57 2.9.2 Operatorinequalitiesforpositivelinearmaps|62 2.10 OnconstantsappearinginrefinedYounginequality|64 3 Inequalitiesrelatedtomeans|69 3.1 Operatorinequalitiesforthreemeans|69 3.2 Heronmeansforpositiveoperators|71 3.3 InequalitiesrelatedtoHeronmean|78 3.4 Somerefinementsofoperatorinequalitiesforpositivelinear maps|82 4 Norminequalitiesandtraceinequalities|91 4.1 Unitarilyinvariantnorminequalities|92 4.2 Traceinequality|100 4.2.1 Traceinequalitiesforproductsofmatrices|103 4.2.2 Conjecturedtraceinequality|108 4.2.3 Belmega–Lasaulce–Debbahinequality|113 VIII | Contents 5 ConvexfunctionsandJenseninequality|117 5.1 AgeneralizedoperatorJenseninequality|117 5.2 Choi–Davis–Jenseninequalitywithoutconvexity|124 5.3 OperatorJensen–Mercerinequality|128 5.4 Inequalitiesforoperatorconcavefunction|138 5.5 BoundsonoperatorJenseninequality|143 5.6 Operatorinequalitiesviageometricconvexity|150 5.6.1 Scalarinequalities|151 5.6.2 Numericalradiusinequalities|154 5.6.3 Matrixnorms|160 5.7 Exponentialinequalitiesforpositivelinearmaps|161 5.8 Jensen’sinequalityforstronglyconvexfunctions|170 6 Reversesforclassicalinequalities|179 6.1 ReversesforcomplementedGolden–Thompsoninequalities|179 6.2 ReversesforoperatorAczélinequality|185 6.3 ReversesforBellmanoperatorinequality|191 7 Applicationstoentropytheory|197 7.1 Someinequalitiesonthequantumentropies|198 7.2 Karamatainequalitiesandapplications|203 7.3 Relativeoperatorentropy|210 7.3.1 Refinedboundsforrelativeoperatorentropies|210 7.3.2 Reversestoinformationmonotonicityinequality|215 8 Miscellaneoustopics|225 8.1 Kantorovichinequality|225 8.1.1 NewoperatorKantorovichinequalities|225 8.1.2 Functionsreversingoperatororder|229 8.1.3 NewrefinedoperatorKantorovichinequality|233 8.2 Skewinformationanduncertaintyrelation|238 8.2.1 Skewinformationanduncertaintyrelation|239 8.2.2 SchrödingeruncertaintyrelationwithWigner–Yanaseskew information|241 Bibliography|247 Index|257 1 Introduction and preliminaries 1.1 Introduction Mathematicalinequalitiesareappliedtomanyfieldsonnaturalscienceandengineer- ingtechnology.Someresearchersinsuchfieldsmayfindtheinequalitiesfrombooks orpaperstosolvetheirproblems,whileothersmaydiscovernewinequalitiesifthey donotfindthemintheliterature.Foralmostallresearchers,mathematicalinequal- itiesarejusttoolstosolvetheirproblemsorimprovetheirresults.Suchmotivations arequitenaturaltodeveloptheirstudies,justbecauseofthebeautyoftheirform.On theotherhand,mathematiciansliketodiscovernewinequalities.Theyoftendonot thinkofitspossibleapplications.Afewyearsago,Iwasaskedforaproofofthefol- lowingbeautifulinequalitybymyacquaintancefromJapan,whenIwasinBudapest foraconference.Heaskedmewhetherthefollowinginequalityisalreadyknown.He needed its proof to establish the nonnegativity of thermodynamic entropy for total systems. z x y y z x x>y>z >0 ⇒ x y z >x y z . Igaveaproofforhisquestionasfollow.Sincetherepresentingfunction(t−1)/logt ismonotoneincreasingfort > 0,wehave(u−1)/logu > (t−1)/logt ⇔ tu−1 > ut−1 foru > t > 1.Puttingst = u,wehaves1tts > stfors,t > 1.Puttingagaint = y/z > 1, s=x/y>1,wehave(y/z)x−y >(x/y)y−z ⇔yx−y+y−z >xy−zzx−y ⇔xzyxzy >xyyzzx. HealsoaskedthisquestiontoseveralJapanesemathematiciansandhereported severalproofsduringmystayinBudapest.Everyproofwasdifferent.Ienjoyedhisre- portandguessedthatsodidtheothercolleagues.Afewmathematiciansfoundand proveditsextensionston-variables.Aftermyreturn,Icheckedmybooksonmathe- maticalinequalities,butIcouldnotfindtheliterature.Severalmathematiciansalso couldnotfindtheoriginalsourceofthisinequality.Nevertheless,itisnaturaltocon- siderthatthisinequalityisalreadyknownsinceitisverysimpleandbeautifulinits form.Halfayearlater,oneofours(H.Ohtsuka)foundthatthisinequalitywasposed byM.S.KlamkinandL.A.Sheppin[132]inamoregeneralcase(n-variables): xn ≥xn−1 ≥⋅⋅⋅x2 ≥x1 ≥0⇒x1x2x2x3⋅⋅⋅xnx1 ≥x2x1x3x2⋅⋅⋅x1xn, (n≥3) withequalityholdingonlyifn−1ofnumbersareequal. We leave the solution of this question to the readers. Mathematicians studying inequalitiesareoftenobsessedwithabeautifulinequality.Withinjustoneweekafter thequestion,severalmathematiciansgavedifferentproofsofittoeachother. Wethinktheadvancesofmathematicalinequalitiesmaydependonfindingap- plicationstootherfieldsand/orinterestsininequalitiesthemselves.Here,wepresent ourrecentadvancesonmathematicalinequalities.Almostallresultswereestablished withinthelastfewyears.However,wehaveaddedlessrecentresultssincetheymay https://doi.org/10.1515/9783110643473-001