Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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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>). | 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: | We have: | ||
− | <math>F\left(\frac{x_1}{n}+\cdots+\frac{x_n}{n}\right)\le \ | + | <math>F\left(\frac{x_1}{n}+\cdots+\frac{x_n}{n}\right)\le \frac1)}{n}+\cdots+\frac{F(x_n)}{n}</math>; |
Factoring out the <math>\frac{1}{n}</math> yields: | Factoring out the <math>\frac{1}{n}</math> yields: |
Revision as of 19:45, 5 November 2014
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, the RMS-AM can be proved using Jensen's inequality: Suppose we let (We know that is convex because and therefore ). We have: $F\left(\frac{x_1}{n}+\cdots+\frac{x_n}{n}\right)\le \frac1)}{n}+\cdots+\frac{F(x_n)}{n}$ (Error compiling LaTeX. Unknown error_msg);
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.
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