Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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== Proof == | == 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>\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>. | ||
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+ | Alternatively, it can be proved using Jensen's inequality: | ||
+ | Suppose we let <math>F(x)=x^2</math> (We know that <math>F(x)</math> is convex because <math>F'(x)=2x</math> and therefore <math>F''(x)=2>0</math>). | ||
+ | We have: | ||
+ | <math>F(\frac{x_1}{n}+\cdots+\frac{x_n}{n})\le \frac{F(x_1)}{n}+\cdots+\frac{F(x_n)}{n}</math>; | ||
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+ | Factoring out the <math>\frac{1}{n}</math> yields: | ||
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+ | <math>F(\frac{x_1+\cdots+x_n}{n})\le \frac {F(x_1)+\cdots+F(x_n)}{n}</math> | ||
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+ | |||
+ | <math>(\frac{x_1+\cdots+x_n}{n})^2 \le \frac{x_1^2+\cdots+x_n^2}{n}</math> | ||
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+ | Taking the square root to both sides (remember that both are positive): | ||
+ | |||
+ | <math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n} \blacksquare.</math> | ||
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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>\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]]. |
Revision as of 19:43, 9 May 2013
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 .
Alternatively, it 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 know as the quadratic mean, and the inequality is therefore sometimes known as the QM-AM-GM-HM Inequality.
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