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arithmetic-geometric-harmonic means inequality
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(Theorem)
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Let $x_1,x_2,\ldots,x_n$ be positive numbers. Then \begin{eqnarray*} \max\{x_1,x_2,\ldots,x_n\} &\ge& \frac{x_1+x_2+\cdots+x_n}{n}\\ &\ge& \sqrt[n]{x_1 x_2\cdots x_n} \\ &\ge& \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}}\\ &\ge& \min\{x_1,x_2,\ldots,x_n\} \end{eqnarray*} The equality is obtained if and only if $x_1=x_2=\cdots = x_n$
There are several generalizations to this inequality using power means and weighted power means.
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"arithmetic-geometric-harmonic means inequality" is owned by drini. [ owner history (1) ]
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See Also: arithmetic mean, geometric mean, harmonic mean, general means inequality, weighted power mean, power mean, root-mean-square, proof of general means inequality, Jensen's inequality, derivation of geometric mean as the limit of the power mean, minimal and maximal number, proof of arithmetic-geometric means inequality using Lagrange multipliers, comparison of Pythagorean means, Heronian mean is between geometric and arithmetic mean
| Other names: |
harmonic-geometric-arithmetic means inequality, arithmetic-geometric means inequality, AGM inequality, AGMH inequality |
| Keywords: |
inequality, mean, arithmetic mean, geometric mean, harmonic mean |
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Cross-references: weighted power means, power means, inequality, equality, numbers, positive
There are 11 references to this entry.
This is version 5 of arithmetic-geometric-harmonic means inequality, born on 2001-08-18, modified 2004-06-05.
Object id is 25, canonical name is ArithmeticGeometricMeansInequality.
Accessed 34758 times total.
Classification:
| AMS MSC: | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
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Pending Errata and Addenda
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