Advances on Inequalities of the Schwarz, Triangle

arXiv:math/0503059v1[math.FA]3Mar2005AdvancesonInequalitiesoftheSchwarz,TriangleandHeisenbergTypeinInnerProductSpacesSeverSilvestruDragomirSchoolofComputerScience&Mathematics,VictoriaUniversity,Melbourne,Victoria,AustraliaE-mailaddress:[e-mail]:fication.Primary46C05,46E30;Secondary25D15,26D10Abstract.ThepurposeofthisbookistogiveacomprehensiveintroductiontoseveralinequalitiesinInnerProductSpacesthathaveimportantapplica-tionsinvarioustopicsofContemporaryMathematicssuchas:LinearOper-atorsTheory,PartialDifferentialEquations,NonlinearAnalysis,Approxima-tionTheory,OptimizationTheory,NumericalAnalysis,ProbabilityTheory,Statisticsandotherfields.ContentsPrefacevChapter1.InequalitiesforHermitianForms11.1.Introduction11.2.HermitianForms,FundamentalProperties11.3.SuperadditivityandMonotonicity71.4.ApplicationsforGeneralInnerProductSpaces131.5.ApplicationsforSequencesofVectors22Bibliography31Chapter2.SchwarzRelatedInequalities332.1.Introduction332.2.InequalitiesRelatedtoSchwarz’sOne342.3.KurepaTypeRefinementsfortheSchwarzInequality402.4.RefinementsofBuzano’sandKurepa’sInequalities452.5.InequalitiesforOrthornormalFamilies512.6.GeneralizationsofPrecupanu’sInequality572.7.SomeNewRefinementsoftheSchwarzInequality642.8.MoreSchwarzRelatedInequalities75Bibliography89Chapter3.ReversesfortheTriangleInequality913.1.Introduction913.2.SomeInequalitiesofDiaz-MetcalfType923.3.AdditiveReversesfortheTriangleInequality953.4.FurtherAdditiveReverses983.5.ReversesofSchwarzInequality1023.6.QuadraticReversesoftheTriangleInequality1033.7.FurtherQuadraticRefinements1093.8.ReversesforComplexSpaces1143.9.ApplicationsforVector-ValuedIntegralInequalities1203.10.ApplicationsforComplexNumbers123Bibliography125Chapter4.ReversesfortheContinuousTriangleInequality1274.1.Introduction1274.2.MultiplicativeReverses1284.3.SomeAdditiveReverses135iiiivCONTENTS4.4.QuadraticReversesoftheTriangleInequality1454.5.RefinementsforComplexSpaces1514.6.ApplicationsforComplex-ValuedFunctions158Bibliography165Chapter5.ReversesoftheCBSandHeisenbergInequalities1675.1.Introduction1675.2.SomeReverseInequalities1685.3.OtherReverses181Bibliography191Chapter6.OtherInequalitiesinInnerProductSpaces1936.1.BoundsfortheDistancetoFinite-DimensionalSubspaces1936.2.ReversingtheCBSInequalityforSequences2046.3.OtherReversesoftheCBSInequality221Bibliography233PrefaceThepurposeofthisbook,thatcanbeseenasacontinuationoftheprevi-ousoneentitled”AdvancesonInequalitiesoftheSchwarz,Gr¨ussandBesselTypeinInnerProductSpaces”(NovaSciencePublishers,NY,2005),istogiveacom-prehensiveintroductiontootherclassesofinequalitiesinInnerProductSpacesthathaveimportantapplicationsinvarioustopicsofContemporaryMathematicssuchas:LinearOperatorsTheory,PartialDifferentialEquations,NonlinearAnaly-sis,ApproximationTheory,OptimizationTheory,NumericalAnalysis,ProbabilityTheory,Statisticsandotherfields.ThemonographisintendedforusebybothresearchersinvariousfieldsofMathematicalInequalities,domainswhichhavegrownexponentiallyinthelastdecade,aswellasbypostgraduatestudentsandscientistsapplyinginequalitiesintheirspecificareas.TheaimofChapter1istopresentsomefundamentalanalyticpropertiescon-cerningHermitianformsdefinedonrealorcomplexlinearspaces.Thebasicin-equalitiesaswellasvariouspropertiesofsuperadditivityandmonotonicityforthediversefunctionalsthatcanbenaturallyassociatedwiththequantitiesinvolvedintheSchwarzinequalityaregiven.Applicationsfororthonormalfamilies,Gramdeterminants,linearoperatorsdefinedonHilbertspacesandsequencesofvectorsarealsopointedout.InChapter2,classicalandrecentrefinementsandreverseinequalitiesfortheSchwarzandthetriangleinequalitiesarepresented.Furtheron,theinequalitiesobtainedbyBuzano,Richards,PrecupanuandMooreandtheirextensionsandgeneralizationsfororthonormalfamiliesofvectorsinbothrealandcomplexinnerproductspacesareoutlined.RecentresultsconcerningtheclassicalrefinementofSchwarzinequalityduetoKurepaforthecomplexificationofrealinnerproductspacesarealsoreviewed.VariousapplicationsforintegralinequalitiesincludingaversionofHeisenberginequalityforvectorvaluedfunctionsinHilbertspacesareprovidedaswell.TheaimofChapter3istosurveyvariousrecentreversesforthegeneralisedtriangleinequalityinbothitssimpleform,thatarecloselyrelatedtotheDiaz-Metcalfresults,orintheequivalentquadraticformthatmaybebeofinterestintheGeometryofInnerproductSpaces.Applicationsforvectorvaluedintegralinequalitiesandforcomplexnumbersaregivenaswell.Furtheron,inChapter4,somerecentreversesofthecontinuoustrianglein-equalityforBochnerintegrablefunctionswithvaluesinHilbertspacesanddefinedonacompactinterval[a,b]⊂Raresurveyed.ApplicationsforLebesgueinte-grablecomplex-valuedfunctionsthatgeneraliseandextendtheclassicalresultofKaramataareprovidedaswell.vviPREFACEInChapter5somereversesoftheCauchy-Buniakovsky-Schwarzvector-valuedintegralinequalitiesundervariousassumptionsofboundednessforthefunctionsinvolvedaregiven.NaturalapplicationsfortheHeisenberginequalityforvector-valuedfunctionsinHilbertspacesa