Muirhead's inequality is a generalization of the AM-GM inequality. Like the AM-GM inequality, it involves a comparison of symmetric sums of monomials involving several variables. It is often useful in proofs involving inequalities.
Let be nonnegative real numbers. Let be variables. Then
is the sum of terms of the form where runs over the permutations of
Now suppose and are two sequences of nonnegative real numbers satisfying the following conditions:
Then the sequence is said to majorize the sequence
majorizes because and
Suppose majorizes Then, for all nonnegative
From the example in the previous section, if are nonnegative real numbers, then
In the interest of more compact notation, write
Then Muirhead's inequality says that if majorizes then
Since majorizes by Muirhead's inequality. In the symmetric sum for there are permutations of the variables that give each term. For instance, the term is preserved by all the permutations that keep fixed and move around. In the symmetric sum for all of the permutations give the same monomial.
So Muirhead's inequality becomes
which is the AM-GM inequality. So this is a special case of Muirhead's inequality.
Find the minimum value of
as range over positive real numbers.
Multiplying out the numerator gives Note that so the expression is
and by Muirhead's inequality, so the expression is But note that if then the expression evaluates to so the answer is
Muirhead's inequality can sometimes be extended to prove non-homogeneous inequalities:
Suppose are positive real numbers with Show that
The trick is to multiply by which makes the degrees of both sides equal to each other. That is,
This last example is quite a bit harder, but illustrates the power of the technique.
[IMO 1995] For with prove that
Multiply by the product of denominators:
Using on the right side, this turns into
Now the left side has degree and the right side has degree To get the degrees on both sides to match up, we can multiply the right side by Note that So,
Now and both majorize so
and majorizes so
and adding these inequalities gives the result.