# Difference between revisions of "User:Temperal/The Problem Solver's Resource11"

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, if <math>m_2</math> is the quadratic mean, <math>m_1</math> is the arithmetic mean, <math>m_0</math> the geometric mean, and <math>m_{-1}</math> the harmonic mean. | , if <math>m_2</math> is the quadratic mean, <math>m_1</math> is the arithmetic mean, <math>m_0</math> the geometric mean, and <math>m_{-1}</math> the harmonic mean. | ||

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+ | ===RSM-AM-GM-HM Inequality=== | ||

+ | For any positive real numbers <math>x_1,\ldots,x_n</math>: | ||

+ | |||

+ | <math>\sqrt{\frac{x_1^2+\cdots+x_n^2}{n}} \ge\frac{x_1+\cdots+x_n}{n}\ge\sqrt[n]{x_1\cdots x_n}\ge\frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}}</math> | ||

+ | |||

+ | with equality iff <math>x_1=x_2=\cdots=x_n</math>. | ||

===Chebyshev's Inequality=== | ===Chebyshev's Inequality=== |

## Revision as of 11:17, 23 November 2007

## InequalitiesMy favorite topic, saved for last. ## Trivial InequalityFor any real , , with equality iff . ## Arithmetic Mean/Geometric Mean InequalityFor any set of real numbers , with equality iff .
## Cauchy-Schwarz InequalityFor any real numbers and , the following holds:
## Cauchy-Schwarz VariationFor any real numbers and positive real numbers , the following holds: . ## Power Mean InequalityTake a set of functions . Note that does not exist. The geometric mean is . For non-negative real numbers , the following holds: for reals . , if is the quadratic mean, is the arithmetic mean, the geometric mean, and the harmonic mean. ## RSM-AM-GM-HM InequalityFor any positive real numbers :
with equality iff . ## Chebyshev's InequalityGiven real numbers and , we have . ## Minkowski's InequalityGiven real numbers and , the following holds:
## Nesbitt's InequalityFor all positive real numbers , and , the following holds: . ## Schur's inequalityGiven positive real numbers and real , the following holds: . ## Jensen's InequalityFor a convex function and real numbers and , the following holds:
## Holder's InequalityFor positive real numbers , the following holds:
## Muirhead's InequalityFor a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
## Rearrangement InequalityFor any multi sets and , is maximized when is greater than or equal to exactly of the other members of , then is also greater than or equal to exactly of the other members of . ## Newton's InequalityFor non-negative real numbers and the following holds: , with equality exactly iff all are equivalent. ## MacLaurin's InequalityFor non-negative real numbers , and such that , for the following holds:
with equality iff all are equivalent. Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |