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wxcfan123
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2015-04-20 13:19:14
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x^2/(x+y^2) = x - xy^2/(x+y^2)
原不等式等價於:
xy^2/(x+y^2) + yz^2/(y+z^2) + zx^2/(z+x^2) <= 3/2.
因x+y^2 >= 2sqrt(xy^2), 歸結於證明
xsqrt(z) + ysqrt(x) + zsqrt(y) <= 3.
由柯西不等式,
LHS^2 <= (x+y+z)(xz+yx+zy) = 3(xz+yx+zy) <= 9
因xz+yx+zy <= (x+y+z)^2/3.