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. | ||
+ | |||
+ | ===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 12:17, 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 RSM-AM-GM-HM InequalityFor any positive real numbers
with equality iff 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! |