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==<span style="font-size:20px; color: blue;">Inequalities</span>== | ==<span style="font-size:20px; color: blue;">Inequalities</span>== | ||
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[[User:Temperal/The Problem Solver's Resource10|Back to page 10]] | Last page (But also see the | [[User:Temperal/The Problem Solver's Resource10|Back to page 10]] | Last page (But also see the | ||
[[User:Temperal/The Problem Solver's Resource Tips and Tricks|tips and tricks page]], and the | [[User:Temperal/The Problem Solver's Resource Tips and Tricks|tips and tricks page]], and the | ||
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Revision as of 18:22, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 11. |
Inequalities
My favorite topic, saved for last.
Trivial Inequality
For any real , , with equality iff .
Arithmetic Mean/Geometric Mean Inequality
For any set of real numbers , with equality iff .
Cauchy-Schwarz Inequality
For any real numbers and , the following holds:
Cauchy-Schwarz Variation
For any real numbers and positive real numbers , the following holds:
.
Power Mean Inequality
Take a set of functions .
Note that does not exist. The geometric mean is . For non-negative real numbers , the following holds:
for reals .
, if is the quadratic mean, is the arithmetic mean, the geometric mean, and the harmonic mean.
RSM-AM-GM-HM Inequality
For any positive real numbers :
with equality iff .
Chebyshev's Inequality
Given real numbers and , we have
.
Minkowski's Inequality
Given real numbers and , the following holds:
Nesbitt's Inequality
For all positive real numbers , and , the following holds:
.
Schur's inequality
Given positive real numbers and real , the following holds:
.
Jensen's Inequality
For a convex function and real numbers and , the following holds:
Holder's Inequality
For positive real numbers , the following holds:
Muirhead's Inequality
For a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
Rearrangement Inequality
For 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 Inequality
For non-negative real numbers and the following holds:
,
with equality exactly iff all are equivalent.
MacLaurin's Inequality
For 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 methods of proof!