**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.

(Lastnightstar): If such that then

**Proof:** We use the same notations as used in . The condition is equivalent to

Hence the inequality after homogenization becomes

which is

(Lastnightstar): If then

**Proof:** Using the same notations by AM-GM

Thus

Again by we have . Using this we have

which is equivalent to

which is precisely what we want to show. Hence the proof is complete.

Note that for three variables becomes

which is the well-known Vasc inequality.

Let us use the notations used by **Lastnightstar** from here.

Let be nonnegative reals. Let

denote the general arithmetic mean of the symmetric sum of variables.

Then maclaurin’s inequality is equivalent to and Newton’s inequality is equivalent to .

Under this notations the two inequalities in become and respectively.

: Let and , then show that

**Proof:** The inequality is equivalent to

Which is true because by Holder:

Similarly we can prove:

: If , then

Show that for suitable values of .

**Proof:** By AM-GM and Cauchy Schwarz we have

(Lastnightstar): Let and let . Show that

**Proof:** Let us prove for variables. For variables the inequality is equivalent to

which Schur’s inequality of degree.

For variables,

Let us simplify the left side:

Note that

Hence

where

Now let us simplify the right side:

Note that

Therefore

Note that Since it is exactly equivalent to for variables namely , i.e., Schur’s inequality of degree.

Hence we have

Show that

**Proof:** It can be proved similarly like the last one.

To be expanded…..

The full thing is already available in my old blog:

http://www.artofproblemsolving.com/community/c2426h1043311_symmetric_n_variable_inequalities