For positive real numbers x, y, z, we have AM > GM
⇒ \(\frac{x^2+y^2}{2}\) > (x2 . y2)\(\frac12\) ⇒ x2 + y2 > 2xy
⇒ x2 + y2 – xy > xy [∵ x > y ⇒ x + a > y + a ∀ a ∈ R]
⇒ (x + y) (x2 + y2 – xy) > xy (x + y) [∵ x > y ⇒ ax > ay ∀ a > 0]
⇒ (x3 + y3) > xy (x + y) …(i)
Similarly, we can show that,
y3 + z3 > yz (y + z) …(ii)
and z3 + x3 > zx (z + x) …(iii)
Adding (i), (ii) and (iii), we have
⇒ x3 + y3 + y3 + z3 + z3 + x3 > xy (x + y) + yz (y + z) + zx (z + x)
⇒ 2 (x3 + y3 + z3) > xy (x + y) + yz (y + z) + zx (z + x)