Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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<math>(x_1+x_2+\cdots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}\right) \geq n^2</math> with equality if and only if <math>x_1=x_2=\cdots=x_n</math>; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving. | <math>(x_1+x_2+\cdots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}\right) \geq n^2</math> with equality if and only if <math>x_1=x_2=\cdots=x_n</math>; which follows directly by cross multiplication from the AM-HM inequality.This is extremely useful in problem solving. | ||
+ | == Proof == | ||
+ | The inequality <math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}</math> is a direct consequence of the [[Cauchy-Schwarz Inequality]]; <math>(x_1^2+x_2^2+\cdots +x_n^2)(1+1+\cdots +1)\geq (x_1+x_2+\cdots +x_n)^2</math>, so <math>\frac{x_1^2+x_2^2+\cdots +x_n^2}{n}\geq \left(\frac{x_1+x_2+\cdots +x_n}{n}\right)^2</math>, so <math>sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}</math>. | ||
+ | The inequality <math>\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}</math> is called the AM-GM inequality, and proofs can be found [[Proofs of AM-GM|here]]. | ||
+ | The inequality <math>\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math> is a direct consequence of AM-GM; <math>\frac{\sum_{i=1}^{n}\sqrt[n]{\frac{x_1x_2\cdots x_n}{x_i^n}}}{n}\geq 1</math>, so <math>\sqrt[n]{x_1x_2\cdots x_n}\frac{\sum_{i=1}^{n}\frac{1}{x_i}}}{n}\geq 1</math>, so <math>\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math>. | ||
+ | |||
+ | Therefore the original inequality is true. | ||
{{stub}} | {{stub}} | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Theorems]] | [[Category:Theorems]] |
Revision as of 13:39, 21 July 2010
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.
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
.
The inequality is called the AM-GM inequality, and proofs can be found here.
The inequality is a direct consequence of AM-GM;
, so $\sqrt[n]{x_1x_2\cdots x_n}\frac{\sum_{i=1}^{n}\frac{1}{x_i}}}{n}\geq 1$ (Error compiling LaTeX. Unknown error_msg), so
.
Therefore the original inequality is true.
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