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

Nikoislife (talk | contribs) |
Smileapple (talk | contribs) |
||

(4 intermediate revisions by 3 users not shown) | |||

Line 1: | Line 1: | ||

− | 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 number]]s <math>x_1,\ldots,x_n</math> that says: | + | The '''Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (RMS-AM-GM-HM) or '''Quadratic Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (QM-AM-GM-HM), is an [[inequality]] of the [[root-mean square]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: |

<cmath>\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}}</cmath> | <cmath>\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}}</cmath> | ||

− | 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]], and is also known as the Mean Inequality Chain. |

As a consequence we can have the following inequality: | As a consequence we can have the following inequality: |

## Latest revision as of 02:36, 27 November 2021

The **Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality** (RMS-AM-GM-HM) or **Quadratic Mean-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality** (QM-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, and is also known as the Mean Inequality Chain.

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.

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.

## Proof

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

### Geometric Proof

The inequality is clearly shown in this diagram for

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