x>=y>=z>0,求证：x^2*y/z+y^2*z/x+z^2*x/y>=x^2+y^2+z^2
问题描述：
x>=y>=z>0,求证：x^2*y/z+y^2*z/x+z^2*x/y>=x^2+y^2+z^2
答
首先注意如下关系：(x^2y/z+y^2z/x+z^2x/y) - (xy^2/z+yz^2/x+zx^2/y)=(xy/z)(x-y) + (yz/x)(y-z) + (zx/y)(z-x)=(xy/z)(x-y) + (yz/x)(y-z) - (zx/y)(x-y) - (zx/y)(y-z)=(xy/z - zx/y)(x-y) + (yz/x - zx/y)(y-z)=...能否运用排序不等式的有关知识给出证明？