**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

**Proof:** See http://www.artofproblemsolving.com/Wiki/index.php/Newton%27s_Inequality

**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

Hence

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

Then for

And

Hence by induction the two inequalities are true.