Difference between revisions of "Minkowski Inequality"
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Let <math>r>s</math> be a nonzero real number, then for any positive numbers <math>a_{ij}</math>, the following inequality holds: | Let <math>r>s</math> be a nonzero real number, then for any positive numbers <math>a_{ij}</math>, the following inequality holds: | ||
| − | <math>(\sum_{j=1}^{m}(\sum_{i=1}^{n}a_{ij}^r)^{s/r})^{1/s}\geq (\sum_{i=1}^{n}(\sum_{j=1}^{m}a_{ij}^s)^{r/s})^{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 one of <math>r,s</math> is zero, the inequality is | + | Notice that if one of <math>r,s</math> is zero, the inequality is equivalent to [[Holder's Inequality]]. |
| + | == Problems == | ||
| + | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=432791#432791 AIME 1991 Problem 15] | ||
{{wikify}} | {{wikify}} | ||
Revision as of 13:26, 19 December 2007
Minkowski Inequality states:
Let 
be a nonzero real number, then for any positive numbers 
, the following inequality holds:
Notice that if one of 
is zero, the inequality is equivalent to Holder's Inequality.
Problems
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