不等式问题:正实数x,y,z满足xyz≥1,证明(x5-x2)/(x5+y2+z2)+(y5-y2)/(y5+z2+x2

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/08/28 12:45:53

不等式问题:正实数x,y,z满足xyz≥1,证明(x5-x2)/(x5+y2+z2)+(y5-y2)/(y5+z2+x2)+(z5-z2)/(z5+x2+y2)≥0
字母右边的数字是指数,
应该是用柯西不等式的

(x^5-x^2)/(x^5+y^2+z^2)-(x^5-x^2)/(x^3(x^2+y^2+z^2))
=[x^2(x^3-1)^2(y^2+z^2)]/[x^3(x^5+y^2+z^2)(x^2+y^2+z^2)]>=0
所以∑(x^5-x^2)/(x^5+y^2+z^2)
>=∑(x^5-x^2)/(x^3(x^2+y^2+z^2))
=(1/(x^2+y^2+z^2))∑(x^2-1/x)
>=(1/(x^2+y^2+z^2))∑(x^2-yz)>=0
http://tieba.baidu.com/f?kz=559358243. 这里也有一种证明,不过没上面那种好

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