Difference between revisions of "AM-GM Inequality"
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Etmetalakret (talk | contribs) (Moving the proof to the other article; it's just too long to fit here, and a beginning student would get terrified looking at it.) |
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− | In [[Algebra]], the '''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 real in the list is the same. |
In symbols, the inequality states that for any real numbers <math>x_1, x_2, \ldots, x_n \geq 0</math>, <cmath>\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. | In symbols, the inequality states that for any real numbers <math>x_1, x_2, \ldots, x_n \geq 0</math>, <cmath>\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}</cmath> with equality if and only if <math>x_1 = x_2 = \cdots = x_n</math>. | ||
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== Proofs == | == Proofs == | ||
{{Main|Proofs of AM-GM}} | {{Main|Proofs of AM-GM}} | ||
− | All known proofs of AM-GM use either induction or other inequalities. Its proof is far more complicated than its usage in introductory competitions, | + | 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 <math>n</math> implies <math>n-1</math>. |
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== Generalizations == | == Generalizations == | ||
− | The AM-GM Inequality has been generalized into several other inequalities. In addition to those listed, the [[Minkowski Inequality]] | + | 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 === | === Weighted AM-GM Inequality === | ||
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=== Power Mean Inequality === | === Power Mean Inequality === | ||
{{Main|Power Mean Inequality}} | {{Main|Power Mean Inequality}} | ||
− | The '''Power Mean Inequality''' relates every power mean of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = | + | The '''Power Mean Inequality''' relates every power mean of a list of nonnegative reals. The power mean <math>M(p)</math> is defined as follows: <cmath>M(p) = \begin{cases} \left( \frac{x_1^p + x_2^p + \cdots + x_n^p}{n}\right)^\frac{1}{p} &\text{if } p \neq 0 \ \sqrt[n]{x_1 x_2 \cdots x_n} &\text{if } p = 0. \end{cases}</cmath> The Power Mean inequality then states that if <math>a>b</math>, then <math>M(a) \geq M(b)</math>, with equality holding if and only if <math>x_1 = x_2 = \cdots = x_n.</math> Plugging <math>p=1, 0</math> into this inequality reduces it to AM-GM, and <math>p=2, 1, 0, -1</math> gives the Mean Inequality Chain. As with AM-GM, there also exists a weighted version of the Power Mean Inequality. |
== Introductory examples == | == Introductory examples == |
Revision as of 15:06, 19 December 2021
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 real 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
[hide]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 Inequality
The Power Mean Inequality relates every power mean 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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