Arithmetic Mean-Geometric Mean Inequality
The Arithmetic Mean-Geometric Mean Inequality (AM-GM or AMGM) is an elementary inequality, generally one of the first ones taught in inequality courses.
For a set of nonnegative real numbers , the following always holds:
For example, for the set , the Arithmetic Mean, 25, is greater than the Geometric Mean, 18; AM-GM guarantees this is always the case.
The weighted form of AMGM is given by using weighted averages. For example, the weighted arithmetic mean of and with is and the geometric is . AMGM applies to weighted averages.
- The power mean inequality is a useful inequality extending on the arithmetic mean side of the inequality.
- The root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality is an extension on both sides of the inequality with further types of means.
- Find the minimum value of for .
- Let , , and be positive real numbers. Prove that