The Arithmetic Mean - Geometric Mean Inequality states that for any non-negative real values , the arithmetic mean of these numbers is greater than or equal to their geometric mean, with equality holding if and only if all the values are equal, i.e.,
The proof of this statement for two variables is presented in the post Completing the Square.
The proof of the general case is given here.
Here are some consequences of AM-GM:
If is a positive real number, then .
If and are positive real numbers, then .
If is a real number (not necessarily positive), then .
1. Show that for .
Solution: We apply the 3-variable version of AM-GM with and to obtain
Then we multiply both sides by 3 to obtain .
2. Find all real solutions to .
Solution: We have , so . This implies is the only possible value. Since and , we have verified is the only solution.
3. Find all positive real solutions to
Solution: By AM-GM, we have . Summing these three inequalities, we obtain
Furthermore, summing the three given equations, we obtain
Hence, equality must hold throughout, implying and . By substituting these values into the original equations, we see that is indeed a solution.
4. [2-variable Cauchy Schwarz Inequality] Show
Solution 1: Expanding both sides, we can cancel terms and , so we need to show that . This follows from the 2-variable AM-GM by setting and , to obtain
Solution 2: From Completing the Square's Fermat's Two Square Theorem, we have
Since squares are non-negative, the right hand side is greater than or equal to .
5. Show that if , and are positive real numbers, then
Solution: A direct application of AM-GM doesn’t seem to work. Let's consider how we can get terms on the right hand side through AM-GM. To get , we will need 'more' of than of or (as in Worked Example 1). This gives a hint to try
Similarily, we have
Adding these 3 inequalities and dividing by 4 yields