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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 $r$-mean (or $r$-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 $x,y$ such that $xy\neq 0$ and $x<y$, we have

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

and the equality holds if and only if $a_1 = ... = a_n$.

Additionally, if we define $M^0$ to be the geometric mean $(a_1a_2...a_n)^{1/n}$, we have that the inequality above holds for arbitrary real numbers $x<y$.

The mentioned inequality is a special case of this one, since $M^1$ is the arithmetic mean, $M^0$ 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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See Also: arithmetic-geometric-harmonic means inequality, arithmetic mean, geometric mean, harmonic mean, power mean, root-mean-square, derivation of zeroth weighted power mean, proof of arithmetic-geometric-harmonic means inequality, comparison of Pythagorean means

Other names:  power means inequality

Attachments:
proof of general means inequality (Proof) by pbruin
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Cross-references: weighted power means, harmonic mean, arithmetic mean, inequality, geometric mean, equality, real numbers, numbers, power mean, arithmetic-geometric means inequality
There are 2 references to this entry.

This is version 3 of general means inequality, born on 2002-05-23, modified 2002-05-23.
Object id is 2934, canonical name is GeneralMeansInequality.
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Classification:
AMS MSC26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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weighted PM inequality by pbruin on 2002-11-22 17:05:59
I have a proof of the weighted power means inequality, but I'm not sure if there should be a separate entry for the WPM inequality. If so, we could link this page and the WPM inequality to each other, otherwise I can attach the proof to this page. In that case, there should be a definition of the WPM inequality on this page in order for the proof to make sense. Which do you think would be the best?
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