广义均值不等式及其简单应用-王琳

28320156()JournalofSichuanUniversityofScience＆Engineering(NaturalScienceEdition)Vol.28No.3Jun.20152015-04-24(1989-)，，，，，(E-mail)[e-mail]1673-1549201503-0096-05DOI10．11863/j．suse．2015．03．201，21．6100592．614000n，，;，，。，。;;;n;O178A1。。2。、3-4。。1。Hn≤Gn≤An≤QnHnHn=n1x1+1x2+…+1xn=n∑ni=11xiGnGn=nx1x2…x槡n=n∏ni=1x槡iAnAn=x1+x2+…+xnn=∑ni=1xinQnQn=x21+x22+…+x2n槡n=∑ni=1x2i槡n256-8。。、、。9-112.1An≤Qnxi≥0i=12…n∑ni=1x2i槡n≥∑ni=1xinx1=x2=…=xn。∑ni=1x2i槡n≥∑ni=1xin∑ni=1x2in≥∑ni=1xi()n2n－1x21+x22+…+x2n≥2x1x2+2x1x3+…+2xn－1xnnn－1C2n=nn－122xixjx2i+x2ji≠ji=123…nj=123…n。x2i+x2j≥2xixji≠ji=123…nj=123…n。n－1x21+x22+…+x2n≥2x1x2+2x1x3+…+2xn－1xnx1=x2=…=xn。xi≥0i=12…n∑ni=1x2i槡n≥∑ni=1xinx1=x2=…=xn。2.2Hn≤Gn≤Anxi≥0i=12…nn∑ni=11xi≤n∏ni=1x槡i≤∑ni=1xin1x1=x2=…=xn。Ⅰn=2kk=12…。x1x槡2=x1+x2()22－x1－x2()2槡2≤x1+x222x1=x2。4x1x2x3x槡4=x1x槡2x3x槡槡42x1x槡2x3x槡槡4≤x1x槡2+x3x槡42≤x1+x2()2+x3+x4()224x1x2x3x槡4≤x1+x2+x3+x44x1=x2=x3=x4。k∈Nk2kx1x2…x2槡k≤2k－1x1+x22·x3+x42…x2k－1+x2k2槡≤…≤x1+x2+…+x2k2kx1=x2=…=x2k。ⅡX=x1+x2+…+xnnnX=x1+x2+…+xn。n+1X=nX+Xn+1=x1+x2+…+xn+Xn+1≥n+1x1x2…xn槡XXn+1≥x1x2…xnXXn≥x1x2…xnX≥x1x2…xn1nn。n+1x1=x2=…=xn。xi≥0i=12…nnx1x2…x槡n≤x1+x2+…+xnnx1=x2=…=xn。1x11x2…1xnnx1x2…x槡n≥n1x1+1x2+…+1xnx1=x2=…=xn。xi≥0i=12…n79283，:n∑ni=11xi≤n∏ni=1x槡i≤∑ni=1xinx1=x2=…=xn。33.1xi≥0i=12…nMrx≡1n∑nix()ri1r≡xr1+xr2+…+xrn()n1rr＞0Mrxx1x2…xnr。M1x=x1+x2+…+xnn≡AnxMrx=1n∑nix()ri1r≡xr1+xr2+…+xrn()n1r≡Axr1r3.2r＞0fx＞0px＞0abAf=∫bapxfxdx∫bapxdxMrf=∫bapxfrxdx∫bapxdx1rGf=exp∫bapxlnfxdx∫bapxdxGf≤AfGf≤Mrfqx=px∫bapxdxpxe∫baqxlnfxdx≤∫baqxfrxd()x1rr＞0r=1r＞0Mr2f=∫baqxfr2xd()x2r=∫baqx槡qx槡fr2d()x2rSchwarzMr2()f≤∫baqxdx·∫baqxfrxd()x1r∫baqxdx=1Mr2()f≤∫baqxdx·∫baqxfrxd()x1r=∫baqxfrxd()x1r=MrfMr2f≤Mrf3fx＞0qx＞0ababx0∈abfx0=maxa≤x≤bfx=u∫baqxfrxd()x1r≤∫baqxurd()x1r=u∫baqxd()xr→+→∞ufxx0ε＞0αβabfx＞u－εxαβ∫baqxfrd()x1r≥∫βαqxfrd()x1r≥u－ε∫baqxd()x1rr→+→∞u－εε＞0limr→∞∫baqxfrd()x1r=u=maxa≤x≤bfx∫baqxlnfxdx≤lnu∫baqxdxGf=exp∫baqxlnfxdx≤elnu∫baqxdx=u1Mrf≥Mr2f≥Mr4f≥Mr2if≥…limr→0+Mrf=uMrf≥Gfr=1Af≥Gf。89()20156“≥”fx≡cc。4、。1gx01e∫10lngxdx≤∫10gxdxgxlngx01。01n∫10gxdx=limn→∞1n∑ni=1gi()n∫10lngxdx=limn→∞1n∑ni=1lngi()n=limn→∞ln∏ni=1gi()()n1ne∫10lngxdx=elimn→∞ln∏ni=1gin1n=limn→∞∏ni=1gi()()n1n∏ni=1gi()()n1n≤1n∑ni=1gi()ne∫10lngxdx≤∫10gxdx2。。。y0yiiYmnxnn。xnyiy0m+y1xix1m=y0m+y1x1=my0m+y1=y01+y1mxn=y01+y1()m1+y2()m…1+yn()m4∑ni=1yi=Y51+y1()m1+y2()m…1+yn()m≤∑ni=11+yi()m[]nn62、34xn≥y01+Y()nmn1+y1m=1+y2m=…=1+ynm。xny1=y2=…=yn。xn=y01+Y()nmn5n。“”—“”99283，:--------GeneralizedMeanInequalityandItsSimpleApplicationWANGLin1，YANGXiu2(1．SchoolofAdministrativeScience，ChengduUniversityofTechnology，Chengdu610059，China;2．SchoolofMathematicsandInformationScience，LeshanNormalUniversity，Leshan614000，China)Abstract:Themeaninequalityisgeneralizedfromtwo-dimensionalspaceton-dimensionalspace，andthegeneralizedmeaninequalityisprovedthroughthereversederivationmethodandreverseinductionmethod，whichprovesthevalidityofthegeneralapproachesofinequality．Thepowergeneralformandintegralformofgeneralizedmeaninequalityarepresentedinformandtheory，andthemethodofprovingam-gminequalityinintegralisfurtherstudiedthroughthecombinationwithpropertiesofbasicam-gminequality，thenthetheoryapplicationscopeofmeaninequalityisexpanded．Examplesfullyreflectthenaturesofam-gminequalityandhowtocombineitwithmathematicalmodelingthoughttosolveproblems，whichshowsthatthegeneralizedmeaninequalityisveryimportant．Keywords:generalizedmeaninequality;integralform;two-dimensionalspace;n-dimensionalspace;basicaveragein-equality001()20156