Arithmetic Mean-Geometric Mean Inequality
The Arithmetic Mean-Geometric Mean Inequality (AM-GM or AMGM) is an elementary inequality, and is generally one of the first ones taught in inequality courses.
For a set of nonnegative real numbers , the following always holds: Using the shorthand notation for summations and products: For example, for the set , the arithmetic mean, 25, is greater than the geometric mean, 18; AM-GM guarantees this is always the case.
See here: Proofs of AM-GM.
The weighted form of AM-GM is given by using weighted averages. For example, the weighted arithmetic mean of and with is and the geometric is .
AM-GM applies to weighted averages. Specifically, the weighted AM-GM Inequality states that if are nonnegative real numbers, and are nonnegative real numbers (the "weights") which sum to 1, then or, in more compact notation, Equality holds if and only if for all integers such that and . We obtain the unweighted form of AM-GM by setting .
- The power mean inequality is a generalization of AM-GM which places the arithemetic and geometric means on a continuum of different means.
- The root-square-mean arithmetic-mean geometric-mean harmonic-mean inequality is special case of the power mean inequality.
- For nonnegative real numbers , demonstrate that if then . (Solution)
- Find the maximum of for all positive . (Solution)
- Find the minimum value of for .
- Let , , and be positive real numbers. Prove that