Difference between revisions of "User:Temperal/The Problem Solver's Resource11"
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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 be</math>, the following holds: | For positive real numbers <math>a_{i_{j}}, 1\le i\le m, 1\le j\le n be</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> | <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 < | + | 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: |
<cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath> | <cmath>\sum_{sym} {x_1}^{a_1}{x_2}^{a_2}\ldots {x_n}^{a_n}\geq \sum_{sym} {x_1}^{b_1}{x_2}^{b_2}\cdots {x_n}^{b_n}</cmath> | ||
===Rearrangement Inequality=== | ===Rearrangement Inequality=== | ||
− | For any multi sets < | + | For any multi sets <math>{a_1,a_2,a_3\ldots,a_n}</math> and <math>{b_1,b_2,b_3\ldots,b_n}</math>, <math>a_1b_1+a_2b_2+\ldots+a_nb_n</math> is maximized when <math>a_k</math> is greater than or equal to exactly <math>i</math> of the other members of <math>A</math>, then <math>b_k</math> is also greater than or equal to exactly <math>i</math> of the other members of <math>B</math>. |
===Newton's Inequality=== | ===Newton's Inequality=== | ||
− | For non-negative real numbers < | + | For non-negative real numbers <math>x_1,x_2,x_3\ldots,x_n</math> and <math>0 < k < n</math> the following holds: |
<cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>, | <cmath>d_k^2 \ge d_{k-1}d_{k+1}</cmath>, | ||
− | with equality exactly iff all < | + | with equality exactly iff all <math>x_i</math> are equivalent. |
===Mauclarin's Inequality=== | ===Mauclarin's Inequality=== | ||
− | For non-negative real numbers < | + | For non-negative real numbers <math>x_1,x_2,x_3 \ldots, x_n</math>, the following holds: |
<cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath> | <cmath>x_1 \ge \sqrt[2]{x_2} \ge \sqrt[3]{x_3}\ldots \ge \sqrt[n]{x_n}</cmath> | ||
− | with equality iff all </math> | + | with equality iff all <math>x_i</math> are equivalent. |
[[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 | ||
[[User:Temperal/The Problem Solver's Resource Competition|competition]]! | [[User:Temperal/The Problem Solver's Resource Competition|competition]]! | ||
|}<br /><br /> | |}<br /><br /> |
Revision as of 12:25, 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 |