Difference between revisions of "User:Temperal/The Problem Solver's Resource11"
(→Mauclarin's Inequality: update) |
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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 18:38, 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
Holder's InequalityFor positive real numbers
Muirhead's InequalityFor a sequence
Rearrangement InequalityFor any multi sets Newton's InequalityFor non-negative real numbers
with equality exactly iff all Mauclarin's InequalityFor non-negative real numbers
with equality iff all Back to page 10 | Last page (But also see the tips and tricks page, and the competition! |