证明 :x/(y+z)+y/(z+x)+z/(x+y)>=3/2
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证明 :x/(y+z)+y/(z+x)+z/(x+y)>=3/2
其中 x,y,z>0
其中 x,y,z>0
![证明 :x/(y+z)+y/(z+x)+z/(x+y)>=3/2](/uploads/image/z/19950380-44-0.jpg?t=%E8%AF%81%E6%98%8E+%EF%BC%9Ax%2F%28y%2Bz%29%2By%2F%28z%2Bx%29%2Bz%2F%28x%2By%29%3E%3D3%2F2)
x/(y+z)+y/(x+z)+z/(x+y)>=3/2
设S=x+y+z
x/(y+z)+y/(x+z)+z/(x+y)
=S/(y+z)+S/(x+z)+S/(x+y)-3
>=9/[(y+z)/S+(x+z)/S+(y+x)/S]-3
=9/2-3
=3/2
以上不等号是用算术平均>=调和平均,即:a+b+c/3>=3/(1/a+1/b+1/c)
变一下就是a+b+c>=9/(1/a+1/b+1/c)
设S=x+y+z
x/(y+z)+y/(x+z)+z/(x+y)
=S/(y+z)+S/(x+z)+S/(x+y)-3
>=9/[(y+z)/S+(x+z)/S+(y+x)/S]-3
=9/2-3
=3/2
以上不等号是用算术平均>=调和平均,即:a+b+c/3>=3/(1/a+1/b+1/c)
变一下就是a+b+c>=9/(1/a+1/b+1/c)
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