The AM-GM inequality, formally the inequality of arithmetic and geometric means, states that the arithmetic mean of a list of non-negative real numbers is always greater than or equal to the geometric mean of the same list, with the two means equal only when every number in the list is identical. The simplest non-trivial case covers two non-negative numbers.
Facts
StatementFor a list of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean of the same list, with equality holding if and only if every number in the list is equal. 2 Proof YearDates the first published proof of the general n-variable case (Augustin-Louis Cauchy, Cours d'analyse, 1821); the simplest two-variable case was already known in antiquity. Classification
Statement FormCharacterization Theorem 1 Connections
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In Branch
Source AM-GM Inequality (Wikipedia)
Sources
1. Wikipedia: AM-GM inequality
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number).
2. AM-GM Inequality (Wikipedia)
Wikimedia Foundationopening paragraph
states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same
In Branch: Analysis, Lead
the inequality of arithmetic and geometric means, or more briefly the AM-GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean
View the Source 3. Cours d'Analyse (Wikipedia)
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is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821.
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