Difference between revisions of "Useful inequalities"
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(Created page with "Category:Useful inequalities Here is a list of useful inequalities: {| class="wikitable" border="1" |- ! Name ! Inequality |- | Cauchy-Schwarz inequality | For <mat...") |
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{| class="wikitable" border="1" | {| class="wikitable" border="1" | ||
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! Inequality | ! Inequality | ||
+ | ! Name | ||
+ | ! Conditions | ||
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+ | | <math>\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}</math> | ||
| [[Cauchy-Schwarz inequality]] | | [[Cauchy-Schwarz inequality]] | ||
− | | For <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math> | + | | For <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math> |
+ | |- | ||
+ | | <math>\left(\sum^n_{i=1}|a_i+b_i|^p\right)^\frac{1}{p}\le\left(\sum^n_{i=1}|a_i|^p\right)^\frac{1}{p}+\left(\sum^n_{i=1}|b_i|^p\right)^\frac{1}{p}</math> | ||
+ | | [[Minkowski's inequality]] | ||
+ | | For <math>p\ge 1</math> and <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math> | ||
|} | |} |
Latest revision as of 20:15, 21 April 2015
Here is a list of useful inequalities:
Inequality | Name | Conditions |
---|---|---|
[math]\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}[/math] | Cauchy-Schwarz inequality | For [math]a_1,...,a_n,b_1,...,b_n\in\mathbb{R}[/math] |
[math]\left(\sum^n_{i=1}|a_i+b_i|^p\right)^\frac{1}{p}\le\left(\sum^n_{i=1}|a_i|^p\right)^\frac{1}{p}+\left(\sum^n_{i=1}|b_i|^p\right)^\frac{1}{p}[/math] | Minkowski's inequality | For [math]p\ge 1[/math] and [math]a_1,...,a_n,b_1,...,b_n\in\mathbb{R}[/math] |