Difference between revisions of "User:Temperal/The Problem Solver's Resource11"
(→MacLaurin's Inequality: latex) |
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For any real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,...,b_n</math>, the following holds: | For any real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,...,b_n</math>, the following holds: | ||
− | <math>(\sum a_i^2)(\sum b_i^2) \ge (\sum a_ib_i)^2</math> | + | <math>\left(\sum a_i^2\right)\left(\sum b_i^2\right) \ge \left(\sum a_ib_i\right)^2</math> |
====Cauchy-Schwarz Variation==== | ====Cauchy-Schwarz Variation==== | ||
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<math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>. | <math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>. | ||
+ | |||
===Power Mean Inequality=== | ===Power Mean Inequality=== | ||
Revision as of 12:15, 23 November 2007
InequalitiesMy favorite topic, saved for last. Trivial InequalityFor any real Arithmetic Mean/Geometric Mean InequalityFor any set of real numbers
Cauchy-Schwarz InequalityFor any real numbers
Cauchy-Schwarz VariationFor any real numbers
Power Mean InequalityTake a set of functions Note that
, if Chebyshev's InequalityGiven real numbers
Minkowski's InequalityGiven real numbers
Nesbitt's InequalityFor all positive real numbers
Schur's inequalityGiven positive real numbers
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 MacLaurin'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! |