x = a-d、y=b-d、z = c-dとおく
x≧y≧zとして良い

(d/dx)^2( x^6 + y^6 + z^6 + (x-y)^6 + (y-z)^6 + (z-x)^6-3(x^2y2(x-y)^2+y^2z^2(y-z)^2+z^2x^2(z-x)^2) )
=
90 x^4 - 120 x^3 (y + z) + 144 x^2 (y^2 + z^2) - 84 x (y^3 + z^3) + 24 (y^4 + z^4)
≧ 90x^4-120x^3y+40x^2y^2
+ 90x^4-120x^3z+40x^2z^2
+ 24y^4-84x^3y+73.5x^2y^2
+ 24z^4-84x^3z+73.5x^2z^2
≧0
d/dx( x^6 + y^6 + z^6 + (x-y)^6 + (y-z)^6 + (z-x)^6-3(x^2y2(x-y)^2+y^2z^2(y-z)^2+z^2x^2(z-x)^2) )
=
18 x^5 - 30 x^4 (y + z) + 48 x^3 (y^2 + z^2) - 42 x^2 (y^3 + z^3) + 24 x (y^4 + z^4) - 6 y^5 - 6 z^5
x=yのとき
= {12 z^4 (y^2 - y z) + 12 z^2 (y^2 - y z)^2 + 4 (y^2 - y z)^3 + 3 z^6
≧0
x=yのときの( x^6 + y^6 + z^6 + (x-y)^6 + (y-z)^6 + (z-x)^6-3(x^2y2(x-y)^2+y^2z^2(y-z)^2+z^2x^2(z-x)^2) )
=
12 z^4 (y^2 - y z) + 12 z^2 (y^2 - y z)^2 + 4 (y^2 - y z)^3 + 3 z^6
≧0