Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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[[Image:RMS-AM-GM-HM.gif|frame|right|The inequality is clearly shown in this diagram for <math>n=2</math>]] | [[Image:RMS-AM-GM-HM.gif|frame|right|The inequality is clearly shown in this diagram for <math>n=2</math>]] | ||
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+ | As a consequence we can have the following inequality: | ||
+ | If <math>x_1,x_2,\cdots,x_n</math> are positive reals, then | ||
+ | <math>(x_1+x_2+\cdots+x_n)(\frac{1}{x_1}+\frac{1}{x_2}+\cdots \frac{1}{x_n}) \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. | ||
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Revision as of 02:06, 7 January 2009
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.
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