Difference between revisions of "Minkowski Inequality"
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| − | Minkowski Inequality states | + | The '''Minkowski Inequality''' states that if <math>r>s</math> is a nonzero real number, then for any positive numbers <math>a_{ij}</math>, the following holds: |
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<math>\left(\sum_{j=1}^{m}\left(\sum_{i=1}^{n}a_{ij}^r\right)^{s/r}\right)^{1/s}\geq \left(\sum_{i=1}^{n}\left(\sum_{j=1}^{m}a_{ij}^s\right)^{r/s}\right)^{1/r}</math> | <math>\left(\sum_{j=1}^{m}\left(\sum_{i=1}^{n}a_{ij}^r\right)^{s/r}\right)^{1/s}\geq \left(\sum_{i=1}^{n}\left(\sum_{j=1}^{m}a_{ij}^s\right)^{r/s}\right)^{1/r}</math> | ||
| − | Notice that if | + | Notice that if either <math>r</math> or <math>s</math> is zero, the inequality is equivalent to [[Holder's Inequality]]. |
== Problems == | == Problems == | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=432791#432791 AIME 1991 Problem 15] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=432791#432791 AIME 1991 Problem 15] | ||
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{{stub}} | {{stub}} | ||
[[Category:Inequality]] | [[Category:Inequality]] | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
Revision as of 02:23, 21 April 2008
The Minkowski Inequality states that if 
is a nonzero real number, then for any positive numbers 
, the following holds:
Notice that if either 
or 
is zero, the inequality is equivalent to Holder's Inequality.
Problems
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