# Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"

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<math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math> | <math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math> | ||

− | with equality if and only if <math>x_1=x_2=\cdots=x_n</math>. This inequality can be expanded to the [[power mean inequality]]. | + | with equality if and only if <math>x_1=x_2=\cdots=x_n</math>. This inequality can be expanded to the [[power mean inequality with lots of chess and whatnot]]. |

[[Image:RMS-AM-GM-HM.gif|frame|right|The inequality is clearly shown in this diagram for <math>n=2</math>]] | [[Image:RMS-AM-GM-HM.gif|frame|right|The inequality is clearly shown in this diagram for <math>n=2</math>]] |

## Revision as of 19:35, 16 May 2018

The **Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality** (RMS-AM-GM-HM), is an inequality of the root-mean square, arithmetic mean, geometric mean, and harmonic mean of a set of positive real numbers that says:

with equality if and only if . This inequality can be expanded to the power mean inequality with lots of chess and whatnot.

As a consequence we can have the following inequality: If are positive reals, then with equality if and only if ; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving.

## Proof

The inequality is a direct consequence of the Cauchy-Schwarz Inequality; , so , so .

Alternatively, the RMS-AM can be proved using Jensen's inequality: Suppose we let (We know that is convex because and therefore ). We have: ;

Factoring out the yields:

Taking the square root to both sides (remember that both are positive):

The inequality is called the AM-GM inequality, and proofs can be found here.

The inequality is a direct consequence of AM-GM; , so , so .

Therefore the original inequality is true.

The Root Mean Square is also known as the quadratic mean, and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality.

*This article is a stub. Help us out by expanding it.*