For the first inequality, should a, b, c be all non-negative numbers? For example, if a = -2, b = -3, c = -4, LHS would be -99, whereas RHS equals -72, so LHS is not greater than or equal to RHS.
Assuming a, b, c are positive, notice that a3 + b3 + c3 - 3abc = (a+b+c)(a2+b2+c2-ab-bc-ca) =...