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 *x*≤*y*≤*z*. 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.

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Tags: AM-GM Inequality, Arithmetic Mean, Chebyshev's Inequality, Geometric Mean, Math, Monday Math

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April 13, 2009 at 7:43 pm |

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