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.
(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
(Lastnightstar): If then
Proof: Using the same notations by AM-GM
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.
Let us simplify the left side:
Now let us simplify the right side:
Note that Since it is exactly equivalent to for variables namely , i.e., Schur’s inequality of degree.
Hence we have
Proof: It can be proved similarly like the last one.
To be expanded…..
The full thing is already available in my old blog: