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1、Thegeneralizationofmeansinequality,bySORINPU?PAN?1.Tchebishev’sinequality-discreteformIntoanotherpastarticle,IhavepresentationageneralizationofTchebishev’sinequality,usingthemultiplicativemonotonouskernelnotion.nMoreexactly,ifn-tuples~??aa,,?,a?,???bb,,?,b
2、???aresimilarly12n12nordered,thismeansthatwehavetheinequalities?a?a??b?b??0,?ij,?1,,nthenijijtheTchebishev’sinequalitynnnn??????????pi???pabiii????paii???pbii?,?i?1??i?1??i?1??i?1?p?0,??i1,,nn?2,canbeputintheformin1n0?i?1pbaiii?i?1pbaiii?,n1n0?i?1paii?i?1p
3、aiiif,ofcoursewesupposea?0,??i1,n.iThisleadstonTheorem1.1Ifn-tuples~??aa,,?,a?,???bb,,?,b???,a?0,??i1,,nare12n12nisimilarlyorderedandthecomponentsarenotallequal,thentheaplicationf:???,nx?i?1pbaiiifx???,nx?i?1paiiarestrictlyincresing.Proof.Bymathematicalind
4、uctionforn.Ifn?2,weproofthatifa?aandb?b,thenthefunctionf:???,12122xxpba?pba111222f2?x??xxisstrictlyincreasing.Ofcourse,wehavepa?pa1122xxxbpa1?11?pa22??pb2?2?ba1?2pb2?2?b1?f2?x??xx?b1?x,pa11?pa22paa1?12??p2andtheproblemissolvedinthiscase.Wesupposenowthatthe
5、affirmationistruexxpba111???pbannnfn?1?x?forfandweproofthisforf.Wehavef?x???b?,n?1nnxxnpa11???pa2n1?gx??pnwheregx???,inthehypothesisa?max?a?,andthexxni?aa?????aa?1??in1nn?1ninductionisclosed■Weobservethatintheproofweusedexclusivelythenextexponentialfunctio
6、nproperty:xaiifa?athenthefunctionisdecreasing,andthisconducttothemultiplicativeijxajmonotonouskernelnotion.Definition1.1TheapplicationHAX:???0,??,,AX??nonempty,iscalledmultiplicativemonotonouskernel(resp.multiplicativestrictlymonotonous)ifH(a,x)1?a,a?A,a?a
7、,theapplicationx?isdecreasing(resp.strictly1212H(a,x)2decreasing).WecangeneralizethelasttheorembyTheorem1.2IfHAX:???0,??,,AX??isamultiplicativestrictlymonotonous??bbb?12nkernelandn-tuples~??aa,,...,a?and??,,...,?issimilarlyordered,12n?ccc?12n?c?0,??i1,,nan
8、dthecomponentsarenotallequal,thenthefunctionfX:??in?i?1pbHaxii?i,?fx???n?i?1pcHaxii?i,?isstrictlyincreasing.Proof.WecanmakethisbyinductionlikeinTheorem1.1,orwecanusetheCauchy-Binetidentity:nnnn??????????acii?
