Difference between revisions of "AM-GM inequality"

Revision as of 17:23, 27 November 2021

In algebra, the AM-GM inequality, shorthand for the inequality of arithmetic and geometric means, 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 $x_1, x_2, \ldots, x_n \geq 0$ , $\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n},$ with equality if and only if $x_1 = x_2 = \cdots = x_n$ .

NOTE: This article is a work-in-progress and meant to replace the Arithmetic mean-geometric mean inequality article, which is of poor quality.

OUTLINE:

Proofs
- Links to Proofs of AM-GM Inequality

Generalizations
- Weighted AM-GM
- QM-AM-GM-HM (with or without weights)
- Power Mean (with or without weights)

@@ Line 1: / Line 1: @@
-#REDIRECT [[Arithmetic Mean-Geometric Mean Inequality]]
+In [[algebra]], the '''AM-GM inequality''', shorthand for the '''inequality of arithmetic and geometric means''', 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 <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>.
+'''NOTE''': This article is a work-in-progress and meant to replace the [[Arithmetic mean-geometric mean inequality]] article, which is of poor quality.
+OUTLINE:
+* Proofs
+** Links to [[Proofs of AM-GM Inequality]]
+**
+* Generalizations
+** Weighted AM-GM
+** QM-AM-GM-HM (with or without weights)
+** Power Mean (with or without weights)

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Difference between revisions of "AM-GM inequality"

Revision as of 17:23, 27 November 2021