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

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− | 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> | ||

Line 41: | Line 41: | ||

<asy>size(250); | <asy>size(250); | ||

pair O=(0,0),A=(-1,0),B=(0,1),C=(1,0),P=(1/2,0),Q=(1/2,sqrt(3)/2),R=foot(P,Q,O); | pair O=(0,0),A=(-1,0),B=(0,1),C=(1,0),P=(1/2,0),Q=(1/2,sqrt(3)/2),R=foot(P,Q,O); | ||

− | draw(B--O--C--arc(O,C,A)--O--R--P); | + | draw(B--O--C--arc(O,C,A)--O--R--P); rightanglemark(O,P,R); |

draw(O--B,red); | draw(O--B,red); | ||

draw(P--Q,blue); | draw(P--Q,blue); |

## Latest revision as of 00:52, 6 January 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.

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

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