b*k
发帖数: 27 | 1 Let a, b, c be positive real numbers such that abc = 1.
Prove that
(a-1+1/b)(b-1+1/c)(c-1+1/a) <= 1 | u**x
发帖数: 45 | 2
Consider LHS(Left Hand Side) of above inequality:
LHS=c/b*(1/c-b+1)*(b-1+1/c)*(1-1/c+b)
Let x=(1/c-b+1), y=(b-1+1/c), z=(1-1/c+b).
Since the condition of x,y,z being negative/zero is one of 1/c, b, 1 is larger
than/equal to the sum of the other two, there can be only 2 cases:
1) only one of x, y, z is negative or 0 , obviously LHS<=0<1
2) x>0, y>0, z>0, we have
1/c=(x+y)/2, b=(y+z)/2, 1=(x+z)/2
8xyz
LHS=-------------- (*)
(x+y)(y+z)(z+x)
since x+y>=2(xy)^0.5, y+z>=2(yz)^0.5, z
【在 b*k 的大作中提到】
: Let a, b, c be positive real numbers such that abc = 1.
: Prove that
: (a-1+1/b)(b-1+1/c)(c-1+1/a) <= 1
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