Arithmetic-geometric mean inequality

For nonnegative

a_{1}, \dots, a_{n}

,

\frac{a _{1} + \dots + a _{n}}{n} \geq (a_{1} \dots a_{n})^{1/ n},

with equality exactly when all the

a_{i}

are equal.

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- Weighted arithmetic-geometric mean inequality Arithmetic-geometric mean inequality

Weighted arithmetic-geometric mean inequality ( $\prod_{i} a_{i}^{w_{i}} \leq \sum_{i} w_{i} a_{i}$ )

For nonnegative real numbers

a_{i}

and positive weights

w_{i}

with

\sum_{i} w_{i} = 1

, the weighted geometric mean does not exceed the weighted arithmetic mean:

\prod_{i} a_{i}^{w_{i}} \leq \sum_{i} w_{i} a_{i} .

For positive

a_{i}

, this follows from Jensen's inequality and the concavity of the natural logarithm; a limit extends it to zero inputs. For two terms it gives

a^{s} b^{1 - s} \leq s a + (1 - s) b

when

0 < s < 1

.

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