# Difference between revisions of "User:Temperal/The Problem Solver's Resource11"

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| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 11}} | | style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 11}} | ||

− | ==<span style="font-size:20px; color: blue;"> | + | ==<span style="font-size:20px; color: blue;">Inequalities</span>== |

− | + | My favorite topic, saved for last. | |

+ | ===Trivial Inequality=== | ||

+ | For any real <math>x</math>, <math>x^2\ge 0</math>, with equality iff <math>x=0</math>. | ||

+ | ===Arithmetic Mean/Geometric Mean Inequality=== | ||

+ | For any set of real numbers <math>S</math>, <math>\frac{S_1+S_2+S_3....+S_{k-1}+S_k}{k}\ge \sqrt[k]{S_1\cdot S_2 \cdot S_3....\cdot S_{k-1}\cdot S_k}</math> with equality iff <math>S_1=S_2=S_3...=S_{k-1}=S_k</math>. | ||

+ | |||

+ | |||

+ | ===Cauchy-Schwarz Inequality=== | ||

+ | |||

+ | 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> | ||

+ | |||

+ | ====Cauchy-Schwarz Variation==== | ||

+ | |||

+ | For any real numbers <math>a_1,a_2,...,a_n</math> and positive real numbers <math>b_1,b_2,...,b_n</math>, the following holds: | ||

+ | |||

+ | <math>\sum\left({{a_i^2}\over{b_i}}\right) \ge {{\sum a_i^2}\over{\sum b_i}}</math>. | ||

+ | ===Power Mean Inequality=== | ||

+ | |||

+ | Take a set of functions <math>m_j(a) = \left({\frac{\sum a_i^j}{n}}\right)^{1/j}</math>. | ||

+ | |||

+ | Note that <math>m_0</math> does not exist. The geometric mean is <math>m_0 = \lim_{k \to 0} m_k</math>. | ||

+ | For non-negative real numbers <math>a_1,a_2,\ldots,a_n</math>, the following holds: | ||

+ | |||

+ | <math>m_x \le m_y</math> for reals <math>x<y</math>. | ||

+ | |||

+ | , 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. | ||

+ | |||

+ | ===Chebyshev's Inequality=== | ||

+ | |||

+ | Given real numbers <math>a_1 \ge a_2 \ge ... \ge a_n \ge 0</math> and <math>b_1 \ge b_2 \ge ... \ge b_n</math>, we have | ||

+ | |||

+ | <math>{\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}</math>. | ||

+ | |||

+ | ===Minkowski's Inequality=== | ||

+ | |||

+ | Given real numbers <math>a_1,a_2,...,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following holds: | ||

+ | |||

+ | <math>\sqrt{\sum a_i^2} + \sqrt{\sum b_i^2} \ge \sqrt{\sum (a_i+b_i)^2}</math> | ||

+ | |||

+ | ===Nesbitt's Inequality=== | ||

+ | |||

+ | For all positive real numbers <math>a</math>, <math>b</math> and <math>c</math>, the following holds: | ||

+ | |||

+ | <math>{\frac{a}{b+c}} + {\frac{b}{c+a}} + {\frac{c}{a+b}} \ge {\frac{3}{2}}</math>. | ||

+ | |||

+ | ===Schur's inequality=== | ||

+ | |||

+ | Given positive real numbers <math>a,b,c</math> and real <math>r</math>, the following holds: | ||

+ | |||

+ | <math>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b)\ge 0</math>. | ||

+ | |||

===Jensen's Inequality=== | ===Jensen's Inequality=== | ||

For a convex function <math>f(x)</math> and real numbers <math>a_1,a_2,a_3,a_4\ldots,a_n</math> and <math>x_1,x_2,x_3,x_4\ldots,x_n</math>, the following holds: | For a convex function <math>f(x)</math> and real numbers <math>a_1,a_2,a_3,a_4\ldots,a_n</math> and <math>x_1,x_2,x_3,x_4\ldots,x_n</math>, the following holds: |

## Revision as of 11:08, 23 November 2007

## InequalitiesMy favorite topic, saved for last. ## Trivial InequalityFor any real , , with equality iff . ## Arithmetic Mean/Geometric Mean InequalityFor any set of real numbers , with equality iff .
## Cauchy-Schwarz InequalityFor any real numbers and , the following holds:
## Cauchy-Schwarz VariationFor any real numbers and positive real numbers , the following holds: . ## Power Mean InequalityTake 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. ## Chebyshev's InequalityGiven real numbers and , we have . ## Minkowski's InequalityGiven real numbers and , the following holds:
## Nesbitt's InequalityFor all positive real numbers , and , the following holds: . ## Schur's inequalityGiven positive real numbers and real , the following holds: . ## Jensen's InequalityFor a convex function and real numbers and , the following holds:
## Holder's InequalityFor positive real numbers , the following holds:
## Muirhead's InequalityFor a sequence that majorizes a sequence , then given a set of positive integers , the following holds:
## Rearrangement InequalityFor 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 InequalityFor non-negative real numbers and the following holds: , with equality exactly iff all are equivalent. ## MacLaurin's InequalityFor 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 competition! |