Difference between revisions of "Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality"
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As a consequence we can have the following inequality: | As a consequence we can have the following inequality: | ||
If <math>x_1,x_2,\cdots,x_n</math> are positive reals, then | 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. | + | <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. |
Revision as of 12:26, 29 April 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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