Monday Math 66

Today, I present a pair of useful inequalities:

1. Chebyshev’s inequality:
Given two sequences of real numbers  and , with at least one of them consisting entirely of positive numbers,
A. If  and  then
,
(the product of the averages is less than or equal to the average of the products).
B. If  and  then
,
(the product of the averages is greater than or equal to the average of the products).
In both cases, equality holds if and only if either  or . (A proof may be found here.)

2. The inequality of arithmetic and geometric means (AM-GM inequality):
For any list of non-negative real numbers, the arithmetic mean is greater than of equal to the geometric mean, with equality only when all of the numbers are the same:
For ,
, with equality only when .
(Several proofs may be found here.)

For an example of usage, consider trying to prove that for x,y,z>0,


First, we note that . Now, due to the symmetry of the above (with repect to permutation of x, y, z), we can assume without loss of generality that xyz. Then it is obvious that , and thus by Chebyshev’s inequality,

and so


Now let us consider the second term in the right-hand-side product:
, and the AM-GM inequality tells us:
.
So ,
and so
.
Q.E.D.

3 Responses to “Monday Math 66”

1. Monday Math 67 « Twisted One 151’s Weblog Says:

[…] numbers with x+y+z=1. Then find (without calculus) the minimum value of . Here, we can use the AM-GM inequality: and , but , so , and so and so , with equality when . Possibly related posts: (automatically […]

2. Monday Math 70 « Twisted One 151’s Weblog Says:

[…] circuit problems (resistors or inductors in parallel, capacitors in series). Now, let us apply the AM-GM inequality to the reciprocal of the harmonic mean: , where is the geometric mean. Thus, . Therefore, for any […]

3. Monday Math 71 « Twisted One 151’s Weblog Says:

[…] . Due to (permutation) symmetry, we may assume without loss of generality, that . Then , and so by Chebyshev’s inequality: And by Nesbitt’s inequality, , […]