试解

令 a=x/z, b=y/x, c=z/y, 则 abc = 1, 并

LHS = ca^3/(ca^2+1) + ab^3/(ab^2+1) + bc^3/(bc^2+1)

= (a+b+c) - (a/(ca^2+1) + b/(ab^2+1) c/(bc^2+1))

ca^2+1 >= 2a*sqrt(c)

>= (a+b+c) - (1/2)(1/sqrt(c) + 1/sqrt(a) + 1/sqrt(b))

= (a+b+c) - (1/2)(sqrt(ab) + sqrt(bc) + sqrt(ca))

a + b + c >= sqrt(ab) + sqrt(bc) + sqrt(ca)

>= (a+b+c)/2 >= (3/2)(abc)^(1/3) = 3/2.