WLG. 可设 x>=y>=z, 则 x+y>=x+z>=y+z
由重排不等式
L >= 1/x(y+z) + 1/y(z+x) + 1/z(x+y) = U.
现有
L+U=(1/(x+y))(1/y+1/z) + (1/(y+z))(1/z+1/x) + (1/(z+x))(1/x+1/y)
= (y+z)/yz(x+y) + (z+x)/zx(y+z) + (x+y)/xy(z+x)
由AG不等式和xyz=1
>=3
故
L >= 3/2
由重排不等式
L >= 1/x(y+z) + 1/y(z+x) + 1/z(x+y) = U.
现有
L+U=(1/(x+y))(1/y+1/z) + (1/(y+z))(1/z+1/x) + (1/(z+x))(1/x+1/y)
= (y+z)/yz(x+y) + (z+x)/zx(y+z) + (x+y)/xy(z+x)
由AG不等式和xyz=1
>=3
故
L >= 3/2
选择“Disable on www.wenxuecity.com”
选择“don't run on pages on this domain”
