Cauchy不等式的不同形式及其证明

Cauchy*潘群星张良云(南京农业大学理学院南京210095)nRn,l2(R),Riemann,(,F,P)Cauchy.Cauchy;Riemann;;O178Cauchy,.Cauchy,.Cauchy.1nRnCauchyni=1aibi2ni=1a2ini=1b2i.(1)1f(x)=ni=1(xai-bi)2!0,(x∀R),x2ni=1a2i-2xni=1aibi+ni=1b2i!0,x∀R.4ni=1aibi2-4ni=1a2ini=1b2i0(1).2(1),n-1i=1n-1j=iaibj+1-aj+1bi2!0..,aibj-ajbi=0(i#ji,,j=1,2,∃n),aibi.3ni=1aia21+∃+a2n2+bib21+∃+b2n2=2!ni=12|aibi|a21+∃+a2nb21+∃+b2n.,ni=1|aibi|2ni=1a2ini=1b2i,(1).2l2(R)Cauchy%n=1a2n%n=1b2n%n=1anbn(),%n=1anbn2%n=1a2n%n=1b2n.(2)109Vol.10,No.4Jul.,2007STUDIESINCOLLEGEMATHEMATICS*:2005-06-25.:&∋.0|anbn|(a2n+b2n)/2%n=1anbn().Sn=ni=1aibi,An=ni=1a2i,Bn=ni=1b2i.(1)S2nAnBn.(2).%n=1a2n%n=1b2n,%n=1(an+bn)2.%n=1(an+bn)212%n=1a2n12+%n=1b2n12.%n=1(an+bn)2.%n=1(an+bn)212%n=1a2n12+%n=1b2n12,%n=1anbn%n=1a2n12%n=1b2n12.,(2)..1%n=1an%n=1bn,.,%n=1an%n=1bn,%n=1(an+bn),limn(%(an+bn)=0.n0an+bn1,(an+bn)2an+bn.%n=1(an+bn)2.3RiemannCauchyf(x)g(x)[a,b]Riemann,)baf(x)g(x)dx2)baf2(x)dx)bag2(x)dx.(3)1[a,b]n,xi=a+b-ani,i=0,1,2,∃n.)baf(x)g(x)dx=limn(%ni=1f(xi)g(xi)b-an,)baf2(x)dx=limn(%ni=1f2(xi)b-an,)bag2(x)dx=limn(%ni=1g2(xi)b-an.(1)ni=1f(xi)g(xi)2ni=1f2(xi)ni=1g2(xi).(3).2F(t)=(tf(x)-g(x))2!0,t2)baf2(x)dx-2t)baf(x)g(x)dx+)bag2(x)dx!0.t∀R,4)baf(x)g(x)dx2-4)baf2(x)dx)bag2(x)dx0(3).3F(t)=)taf(x)g(x)dx2-)taf2(x)dx)tag2(x)dx,(atb),11020077F∗(t)=2f(t)g(t))taf(x)g(x)dx-f2(t))tag2(x)dx-g2(t))taf2(x)dx=-)ta(f(t)g(x)-g(t)f(x))2dx0.,F(t)[a,b],F(b)F(a)=0.(3).2x∀[a,b],f(x)+0f(x)g(x),(3)..,f(x)=g(x),x∀[a,b],)baf(x)dx=)bag(x)dx,f(x)g(x).f(x)g(x)[a,b]Riemann,)ba(f(x)+g(x))2dx12)baf2(x)dx12+)bag2(x)dx12.)baf(x)g(x)dx)baf2(x)dx12)bag2(x)dx12.(3).4(,F,P)CauchyE(X)X.XY(E(XY))2E(X2)E(Y2).(4)t0∀R,P{y=t0X}=1.F(t)=E((tX-Y)2)=t2E(X2)-2tE(XY)+E(Y2)!0.t∀R,4(E(XY))2-4E(X2)E(Y2)0(4).0,t0∀RF(t0)=E((t0X-Y)2)=0.Z=t0X-Y,ZFZ(z).P{|Z|0}=0.P{|Z|0}0,k,P|Z|1k=0.E(Z2)=)+%-%z2dFZ(z)!)|z|1/kz2dFZ(z)1k2)|z|1/kdFZ(z)=1k2P|z|1k=k20.E(Z2)=0.P{|Z|0}=0,P{Y=t0X}=1.5Cauchy!|(,!)||||!|.(5)!...3,(,!)=!T!n,(5)(1).−(f(x),g(x))=)baf(x)g(x)dx,f(x)g(x)[a,b],(5)(3)..(X,Y)=E(XY),XY(,F,P),(5)(4).[1].[M].:,2003.[2].[M].:,1980.[3].[M].:,1999.111104,:Cauchy