The method is a useful technique for proving inequalities involving symmetric polynomials in three real nonnegative variables. These types of problems are common in contest mathematics. The basic idea is to introduce a specific change of variables that simplifies the original inequality.
Let be nonnegative real numbers. The method refers to the following substitutions:
Note that the right sides of these substitutions are the three elementary symmetric polynomials in so every symmetric polynomial in can be written as a polynomial in
The method is commonly used to show that the maximum or minimum value of an expression involving nonnegative real numbers is attained when two of the variables are equal or one of the variables is zero. This requires some important facts about the expressions which are given in the following sections.
The first fact about the change of variables is a simple inequality involving This basic fact motivates the particular choice of variables for the method.
(Basic Theorem) If are nonnegative and are the substitutions in the method, then with equality holding if and only if or (for ) if two of are
First, so with equality if and only if i.e.
Second, is the arithmetic mean of so by AM-GM it is the geometric mean of which is So so and equality holds if and only if which only happens if or two of the variables are
Here is an easy example that uses only the basic theorem.
Suppose are positive real numbers such that Find the minimum value of
After clearing denominators, the constraint becomes or Now Since it is clear that so but since canceling gives So
Finally, the value is attained when so is in fact the minimum value of
Using the method in this way is not particularly powerful: it consists of some simplifying substitutions and a translation of two very basic inequalities (the ones used to prove the basic theorem). The real power of the method comes from a general result known as Tejs' theorem, which is stated and proved below. Tejs' theorem is often used to provide a rigorous justification for a common shortcut: it shows that under certain circumstances, the maximum or minimum of a symmetric expression in three nonnegative real variables occurs when two of the variables are equal, or one of the variables is 0. (In fact, the theorem can be used in more complicated cases, in which more types of triples must also be checked; for an example, see the last section of the wiki.)
When making a change of variables in general, it is often quite important to understand its inverse. In this particular case, passing from the variables to is relatively easy to understand. But suppose we are given (nonnegative) values of and we wish to recover the corresponding This is a more delicate process, and some values of may not even correspond to real values of The following lemma gives conditions that are necessary and sufficient for values of to correspond to real values of
The real numbers satisfy the equations for some real numbers if and only if where Moreover, the following holds:
This is an application of the cubic discriminant and Vieta's formulas. The key fact is that satisfy the equations given in the statement if and only if the polynomial has three real roots. A cubic polynomial has three real roots if and only if its discriminant is nonnegative. Writing in terms of gives which is nonnegative if and only if the quantity in parentheses is nonnegative. This quantity is
As for the statement about nonnegativity, it's clear that if are nonnegative, then are, and is (because it's a product of squares of real numbers). For the converse, suppose are all nonnegative. Proceed by contradiction: if are not all nonnegative, then at least one is negative. Suppose without loss of generality that Since is nonnegative, at least one of them is positive, say without loss of generality that Note also that must be positive. Since is nonnegative, But then is the sum of a negative number and a number that is so it is negative. This is a contradiction, so are all nonnegative.
This lemma will be useful in the proof of Tejs' theorem below. Also, the lemma motivates some notation which will be used in the rest of this wiki.
A triple of nonnegative real numbers will be called admissible if there exist nonnegative real numbers such that
So the lemma says that is admissible if and only if
The idea behind Tejs' theorem is to write a polynomial expression in as an expression in and then to ask where the maximum or minimum of this expression occurs. In particular if two of the three variables are fixed, the expression becomes a polynomial in the third variable, and the maximum/minimum computation becomes a single-variable optimization problem.
The algebra involved can be unpleasant, and the polynomial condition on (from the previous section) is unwieldy, so this is not often a practical way to solve an inequality explicitly. The striking conclusion of Tejs' theorem is that under certain conditions, this technique always yields a maximum or minimum in two very specific cases: either two of are equal or one of is zero. So solving an inequality using Tejs' theorem usually boils down to verifying that the "certain conditions" hold, and then verifying the inequality when or
(1) Suppose are fixed nonnegative real numbers. Let be the set of nonnegative real numbers such that is admissible. Then if is nonempty, it has both minimum and maximum elements, and these elements correspond to triples for which at least two of the variables are equal, or (possibly for the minimum element) one of the variables is zero.
(2) Suppose are fixed nonnegative real numbers. Let be the set of nonnegative real numbers such that is admissible. Then if is nonempty, it has both minimum and maximum elements, and these elements correspond to triples for which at least two of the variables are equal.
(3) Suppose are fixed nonnegative real numbers, and suppose Let be the set of nonnegative real numbers such that is admissible. Then if is nonempty, it has both minimum and maximum elements, and these elements correspond to triples for which at least two of the variables are equal.
(1) The condition that is admissible is i.e. With fixed, this is a quadratic polynomial in say The graph of is a parabola pointing down. If is empty, there is nothing to prove, so assume it is nonempty. Then we seek the set of nonnegative for which is nonnegative. The set of points for which is nonnegative is an interval Note that because otherwise there would be no nonnegative for which was nonnegative, so would be empty. There are two cases.
If the endpoints and are the min and max values of and they correspond to the values of for which is zero. But if and only if which happens if and only if at least two of are equal. So the max and min occur at values where two variables are equal.
If the minimum value of is So so one of the variables is zero. The max still corresponds to two of being the same, by the same argument as the previous paragraph.
(2) Now we search for nonnegative values of such that This time is a cubic polynomial in say Note that is negative for large enough because its lead term is Again, the point is that the set of nonnegative values of for which is nonnegative is, if it is nonempty, a finite union of closed intervals, whose endpoints are either or values of for which The maximum element of this set will be a value of such that which corresponds to two of being equal, and the minimum will either be another value of for which or But if then and since are nonnegative, this is only possible if which only happens if at least two of are zero (hence equal).
(3) Now we search for nonnegative values of such that If this is a cubic in whose lead term is which is negative. The same argument as in the previous paragraph applies. (In this case, corresponds to which implies that )
The theorem has the following two immediate corollaries, which give tools to solve many problems involving symmetric inequalities in three variables:
(1) Any symmetric polynomial of degree in nonnegative real variables with a global minimum and/or maximum will attain this value at triples with either two of the variables equal or one of the variables equal to zero.
(2) Let be a symmetric polynomial of degree in nonnegative real variables Write as where are functions of Then a global minimum and/or maximum of if it exists, will be attained at triples with either two of the variables equal or one of the variables equal to zero, or in the places corresponding to solutions of
For (1), fix and consider the resulting polynomial in ; it is either constant or linear. So its extrema occur when is maximized or minimized, which happens only at the special triples given in part (1) of Tejs' theorem.
For (2), do the same and notice that the resulting polynomial is at most quadratic in So its extrema will occur either at points where is maximized or minimized (i.e. at the endpoints of the domain) or when the quadratic polynomial is at a critical point. Such critical points correspond to places where by elementary calculus. (The point is that this equation might be easier to analyze than the original one, since it has lower degree.)
Let be nonnegative real numbers such that Show that
Consider the quantity This is a symmetric polynomial of degree 3, namely It's linear in with a positive linear coefficient, so it is minimized when is 0 or when is minimal subject to the admissibility constraint, which happens when two variables are the same or one is zero by Tejs' theorem. So it suffices to show the inequality for those particular cases.
When (wlog) we get and our quantity is When are nonnegative, implies (by AM-GM), so Note that this minimum is attained when
When (wlog) we get and our quantity is Substituting gives the quantity as The second factor is always positive, and the first factor is positive on the interval which is where is constrained to lie because and are nonnegative. So the quantity is always in this case as well.
Let be nonnegative real numbers such that Find the minimum value of
Setting and gives We will show that in fact this is the minimum:
To do this, note that this inequality is equivalent to The left side is a symmetric polynomial in of degree In terms of it is which equals in our case (since ). For fixed this is a linear polynomial in which will attain its minimum value either when or when is minimized subject to the admissibility constraint, which happens when two of the variables are equal or one of the variables is zero. So it is enough to look for the minimum value of this expression when two of the variables are equal or one of the variables is zero.
When one of the variables is zero, without loss of generality and The expression becomes where and so this factors as But when clearly (by AM-GM), so the expression is always nonnegative, and attains its minimum value of when
When two of the variables are equal, without loss of generality, Then rewriting in terms of (and using the factorization of in the previous paragraph) gives the constraint as and the expression as and the goal is to show that this is always Making the substitution gives which factors as and each of the three factors on top (as well as the denominator) are always nonnegative for this interval is forced by the constraint , so the expression is
As the previous example shows, the method may still require quite a lot of computation after applying Tejs' theorem.
It is tempting to conclude that Tejs' theorem shows that any inequality involving a symmetric polynomial in three variables need only be checked when two of the variables are equal or one of the variables is zero. This is wrong, as the following example shows:
Let be nonnegative real numbers with and Find the maximum value of
Suppose we check only cases where two of the variables are equal or one variable is zero. If we get and so and which is impossible by AM-GM. If we get and so so which factors as This leads to the two solutions and In the first case, so and in the second case, so So we might erroneously conclude that the minimum is and the maximum is but this is incorrect.
In fact it is not hard to see that the possible values of given the constraints are in the closed interval by applying Tejs' theorem, or by looking at the graph of the cubic function for various values of and concluding that it crosses the -axis three times if and only if However, the quantity is clearly maximized when which occurs at neither of the endpoints of the interval. The maximum value is therefore occurring when are the three real roots of the polynomial
This is the situation described in corollary (2) of Tejs' theorem; the maximum of the expression occurs either when two of the variables are equal or when the quadratic polynomial in has a critical point, i.e. or Polynomials in of larger degree will have more critical points, which will be more difficult to compute and will need to be checked more carefully. (This case is extremely easy even compared to the general degree-6 case, because the quadratic in might have coefficients involving and instead of just rational numbers.) This is why the method is best for symmetric polynomials of low degree.