(文献翻译)凹函数的等价定义及其应用_李莎,左兵,汪义瑞

ThenumberoftheconcavefunctionanditsapplicationLiShaleftWangYirui(Departmentofmathematics,AnkangUniversity,Ankang,Shaanxi725000)Abstract:intheuniversitymathematicslearningandconcavefunctionisakindofspecial.Inmathematicalanalysis,theproofofsomeinequalitiesinprobabilitycourse,withaconcavefunctionofthepriceacaninaveryconcise,cleverlyhavetoprovealoud0withaconcavefunctionthatinequalityofthekeyistoconstructafunctiontobeabletosolvetheproblem.Keywords:concavefunctionequivalentconditioninequalityproof1.concavefunctionDefinition:SetftodefinetheintervalIfunction,iftheITwopointson12,xxArbitrarynumber,(0,1)，thereareatleast1212((1))()(1)()fxxfxfxbefbecalledIConcavefunction.．Note:iftheinequalityischangedtostrictinequality,thenthecorrespondingfunctioniscalledthestrictlyconcavefunction.2.ThenumberofconcavefunctionofthenumberofpiecesEquivalentconditionsofthefirstkind2.1Set(),yfxxI，about1,2xxIisthere12122()()()2fxfxxxf2.2Set(),yfxxI，aboutIThreepointsonthemeaningofthelastterm123xxx，thereareatleast21322132()()()()fxfxfxfxxxxx2.3Set(),yfxxI，aboutIThreepointsonthemeaningofthelastterm123xxx，thereareatleast213232213232()()()()()()fxfxfxfxfxfxxxxxxx，Thatis123kkk2.4Set(),yfxxI，about12,,,,nxxxI，have1212()()(()nnxxxfxfxfxfnn）2.5Set(),yfxxI，about121,,,,(1,2,,)(0,1),1iinnixxxIin，Isthere11()())iiiinniifxfxSecondkindsofequivalentconditions(firstderivative)2.6SetfasintervalI0nthefunctionoftheI0nthefunctionofthe12,xx，there21121()()()()fxfxfxxx．2.7SetfasintervalIonthefunctionofthefbyIontheminusfunction.Thirdkindsofequivalentconditions(twoorderderivative)2.8SetfasintervalIonthetwo-orderderivativefunction,theIThereare0,fxxI3．Theimportantapplicationofconcavefunctionininequalityproving3.1thebasisoftheconcavefunctionisusedtodefinethemeaningof1cases．Set01,,0axy，Provethat.1(1)aaxyaxay．Syndrome:when0xor0yWhentheconclusionisself-evident.②Whenthe,0xyWhentheoriginaltypeln(1)lnln[(1)]axayaxay1AlsolnxAsaconcavefunction,dueto21ln()0xx，utilize()0fxasaconcavefunction,thestructureisestablished.3.2theuseoftheconcavefunctionoftheotherpiecesofevidence2.2Case2.()(01)pppababpProof:theinequalityisrewrittenas:()(01)22pppababpAndinvestigatethefunction,()(0,01)pfxxxp，Weknowthatisaconcavefunction(()0)fx,sotoany12,0xxall1212()()()22xxfxfxfthen1212()22pppxxxx，Therefore,theoriginalinequality.Case3.3334sin(0)2xxxxProof:for3334()sinxxfxx,andremember[0,],[,]442IJbe4(0)=()0,()0,(0)0,()0,()0()244ffffffxxIThusknow()0()fxxJ，Thatis,()fxstayIconcavefunction,，()0()fxxI()xI,fortheintervalJthatissimilar.()0()fxxJ，则3334sin(0)2xxxxCase4.2111(1)()kknnnnknxxn12(1,1;(0,1))knaxxxx1,2,,knProof:because1()()kknfxxxstay(0,)concavefunction(()0)fx，So,wehave1211111111()[()]()kkkknnnnnnkkknxxxnnnnxSotheconclusionwasestablished.3.3usesthepropertyofconcavefunctiontoproveintegralinequality.Corollary:setfunctionin[,]abUppercontinuous,in(,)abTwo-ordercanbeguided,ifthereis()0,()fxgxIsinterval[,]cdIntegrablefunctionson，()agxb，sothere11[()](())ddccfgxdxfgxdxdcdc.Case5．()0gx，Proof101()()ngxdxgnProof:by()0gxknowable()gxIsaconcavefunction,，andnxIsapositivefunctionandsatisfies,01nx，sotheinferencecanbeknowntotheinterval[0,1]integrablefunctionson()gx,if0()0gx，then110100()()1()1()gxdxgxdxgxgxdx，knowable1100()()nngxdxgxdx，Chemicalreduction11001()()()nngxdxgxdxgn，topermit．Reference:[1]MathematicalanalysisofMathematicsDepartmentofEastChinaNormalUniversity(thefirstvolume，ThirdEdition)[M]HigherEducationPress，2001．[2]ZhouMinqiang.Mathematicalanalysisexercises(firstvolume)[M]．SciencePress.2010．[3]PeiWenli.Typicalproblemsandmethodsinmathematicalanalysis[M]．HigherEducationPress.1988．[4]ZhangZhusheng.Newmathematicalanalysis[M]．PekingUniversitypress,2001．[5]XuLizhi.Selectionofmethodsandexamplesofmathematicalanalysis[M]．HigherEducationPress.1984．[6]QiuPetai.Mathematicalanalysisoflearningguidance[M]．SciencePress,2004．