Symmetric Sums: Let be real numbers. We define symmetric sum as the coefficient of in the polynomial
and symmetric average is defined by . For example for variables we have
Newton’s Inequality: Let be non-negative reals. Then for all we have
Maclaurin’s Inequality: Let be nonnegative reals. Let their symmetric averages be . Then we have
Proof: We will use induction and Newton’s Inequality. Let us prove that holds. This inequality is equivalent to
which is obvious.
Suppose the chain of inequalities is true upto . Then by applying Newton’s inequality we have
Thus by induction the proof is complete.
(own): Let denote the arithmetic mean of -th degree symmetric sum of (positive) variables. If and for and for we have the following inequalities:
Proof: To prove this we use induction on .
For , both inequalities are equivalent to which is Newton’s inequality. Let us suppose both holds for all
Hence by induction the two inequalities are true.