A generalization of means inequality

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1、Thegeneralizationofmeansinequality,bySORINPUŞPANĂ1.Tchebishev’sinequality-discreteformIntoanotherpastarticle,IhavepresentationageneralizationofTchebishev’sinequality,usingthemultiplicativemonotonouskernelnotion.nMoreexactly,ifn-tuples~aa,,,a,Ä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,Ä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isamultiplicativestrictlymonotonousÄbbb12nkernelandn-tuples~aa,,...,aand,,...,issimilarlyordered,12nÅccc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