(1) x+y+z=0 のとき、…
(2)
xx+yy+zz = S2，xy+yz+zx = t,
とおく。
S2 - t = {(x-y)^2 + (y-z)^2 + (z-x)^2}/2 ≧ 0,
(S2)^3 - t^3 = {(S2)^2 + S2･t +tt}(S2-t)
≧ {(S2)^2 + S2･t - 2tt}(S2-t)
= (S2+2t)(S2-t)^2
= (x+y+z)^2･{(xx+yy+zz) -(xy-yz-zx)}^2
= (x^3+y^3+z^3 -3xyz)^2,
(3)
yはxとzの中間にあるとしてよい。
0 ≦ (x-y)(y-z) ≦ (1/4)(x-z)^2,
xx+yy+zz = (1/2)(x+z)^2 + (1/2)(x-z)^2 + yy ≧ (1/2)(x-z)^2,
(左辺) ≧ (1/8)(x-z)^6 ≧ 2(x-z)^2 {(x-y)(y-z)}^2 = (右辺),
