Difference between revisions of "User:Temperal/The Problem Solver's Resource11"
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===MacLaurin's Inequality=== | ===MacLaurin's Inequality=== | ||
For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, and <math>d_1,d_2,d_3 \ldots, d_n</math> such that | For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, and <math>d_1,d_2,d_3 \ldots, d_n</math> such that | ||
− | <cmath>d_k = \frac{\ | + | <cmath>d_k = \frac{\sum\limits_{ 1\leq i_1 < i_2 < \cdots < i_k \leq n}x_{i_1} x_{i_2} \cdots x_{i_k}}{{n \choose k}}</cmath>, for <math>k\subset [1,n]</math> the following holds: |
<cmath>d_1 \ge \sqrt[2]{d_2} \ge \sqrt[3]{d_3}\ldots \ge \sqrt[n]{d_n}</cmath> | <cmath>d_1 \ge \sqrt[2]{d_2} \ge \sqrt[3]{d_3}\ldots \ge \sqrt[n]{d_n}</cmath> |
Revision as of 12:09, 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! |