AM-GM Inequality
In Algebra, the AM-GM Inequality, also known formally as the Inequality of Arithmetic and Geometric Means or informally as AM-GM, states that the arithmetic mean is greater than or equal to the geometric mean of any list of nonnegative reals; furthermore, equality holds if and only if every number in the list is the same.
In symbols, the inequality states that for any real numbers ,
with equality if and only if
.
NOTE: This article is a work-in-progress and meant to replace the Arithmetic mean-geometric mean inequality article, which is of poor quality.
Contents
Proofs
- Main article: Proofs of AM-GM
All known proofs of AM-GM use either induction or other, more advanced inequalities. Its proof is far more complicated than its usage in introductory competitions; consequentially, learning it is not recommended to students new to proofs. The most elementary proof of AM-GM utilizes Cauchy Induction, a variant of induction that involves proving a result for two, then using induction to prove it for all powers of two, and then a backward step where implies
.
Generalizations
The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the Minkowski Inequality and Muirhead's Inequality are also generalizations of AM-GM.
Weighted AM-GM Inequality
The Weighted AM-GM Inequality relates the weighted arithmetic and geometric means. It states that for any list of weights such that
,
with equality if and only if
. When
, the weighted form is reduced to the AM-GM Inequality. Several proofs of the Weighted AM-GM Inequality can be found in the proofs of AM-GM article.
Mean Inequality Chain
- Main article: Mean Inequality Chain
The Mean Inequality Chain, also called the RMS-AM-GM-HM Inequality, relates the root mean square, arithmetic mean, geometric mean, and harmonic mean of a list of nonnegative reals. In particular, it states that with equality if and only if
. As with AM-GM, there also exists a weighted version of the Mean Inequality Chain.
Power Mean Inequality
- Main article: Power Mean InequalityThe Power Mean Inequality relates all the different power means of a list of nonnegative reals. The power mean
is defined as follows:
The Power Mean inequality then states that if
, then
, with equality holding if and only if
Plugging
into this inequality reduces it to AM-GM, and
gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality.
Introductory examples
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Intermediate examples
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Olympiad examples
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More Problems
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