Question

一个三元代数不等式命题 设x,y,z;,n,m为正实数,当m>n.有:x/[nx+m(y+z)]+y/[ny+m(z+x

Answer 1

命题 设x,y,z;,n,m为正实数,当m>n.有:

x/[nx+m(y+z)]+y/[ny+m(z+x)]+z/[nz+m(x+y)]>=3/(n+2m)

当m=n时恒等成立,当mn]

[m(b+c)-a(m+n)]/[a(n+2m)(m-n)]+[m(c+a)-b(m+n)]/[b(n+2m)(m-n)]+[m(a+b)-c(m+n)]/[c(n+2m)(m-n)]

= {m[(b+c)/a+(c+a)/b+(a+b)/c]-3(m+n)}/[(n+2m)(m-n)]

>=[6m-3(m+n)]/[(n+2m)(m-n)]=3/(n+2m)=右边.

当n>m时

左边=[m(b+c)-a(m+n)]/[a(n+2m)(m-n)]+[m(c+a)-b(m+n)]/[b(n+2m)(m-n)]+[m(a+b)-c(m+n)]/[c(n+2m)(m-n)]

={m[(b+c)/a+(c+a)/b+(a+b)/c]-3(m+n)}/[(n+2m)(m-n)]

={-m[(b+c)/a+(c+a)/b+(a+b)/c]+3(m+n)}/[(n+2m)(n-m)]

=