Question

设x,y,z为正实数,求证 √[(y^2+yz+z^2)(z^2+zx+x^2)]+√[(z^2+zx+x^2)(x^2

Answer 1

证明 由柯西不等式得:

4√[(y^2+yz+z^2)(z^2+zx+x^2)]

=√{[3(y+z)^2+(y-z)^2]*[3(x+z)^2+(x-z)^2]}

>={√[3(y+z)*(z+x)+(y-z)*(x-z)^2]}^2

=4z^2+4xy+2yz+2zx

同理可得:

4√[(z^2+zx+x^2)(x^2+xy+y^2)]>=4x^2+4yz+2xy+2zx

4√[(x^2+xy+y^2)(y^2+yz+z^2)]>=4y^2+4zx+2yz+2xy

上述三式叠加即得所证不等式.