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general means inequality

(Theorem)

The power means inequality is a generalization of arithmetic-geometric means inequality.

If $0\neq r\in\mathbbmss{R}$ , the -mean (or -th power mean) of the nonnegative numbers $a_1,\ldots,a_n$ is defined as

$\begin{displaymath}M^r(a_1,a_2,\ldots,a_n)= \left(\frac{1}{n}\displaystyle{\sum_{k=1}^n a_k^r}\right)^{1/r}\end{displaymath}$

Given real numbers such that $xy\neq 0$ and , we have

$\begin{displaymath}M^x \leq M^y\end{displaymath}$

and the equality holds if and only if

Additionally, if we define to be the geometric mean $(a_1a_2...a_n)^{1/n}$ , we have that the inequality above holds for arbitrary real numbers .

The mentioned inequality is a special case of this one, since is the arithmetic mean, is the geometric mean and $M^{-1}$ is the harmonic mean.

This inequality can be further generalized using weighted power means.

"general means inequality" is owned by drini. [ owner history (1) ]

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power means inequality

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26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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