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

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===Holder's Inequality=== | ===Holder's Inequality=== | ||

− | For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n | + | For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n</math>, the following holds: |

+ | |||

+ | <cmath>\prod_{i=1}^{m}\left(\sum_{j=1}^{n}a_{i_{j}}\right)\ge\left(\sum_{j=1}^{n}\sqrt[m]{\prod_{i=1}^{m}a_{i_{j}}}\right)^{m}</cmath> | ||

− | |||

===Muirhead's Inequality=== | ===Muirhead's Inequality=== | ||

For a sequence <math>A</math> that majorizes a sequence <math>B</math>, then given a set of positive integers <math>x_1,x_2,\ldots,x_n</math>, the following holds: | For a sequence <math>A</math> that majorizes a sequence <math>B</math>, then given a set of positive integers <math>x_1,x_2,\ldots,x_n</math>, the following holds: |

## Revision as of 17:38, 13 October 2007

## Advanced Number TheoryThese are Olympiad-level theorems and properties of numbers that are routinely used on the IMO and other such competitions. ## 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. ## Mauclarin'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! |