Difference between revisions of "Minkowski Inequality"
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| − | Minkowski Inequality states: | + | The '''Minkowski Inequality''' states that if <math>r>s</math> are nonzero real numbers, then for any positive numbers <math>a_{ij}</math> the following holds: |
| + | <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 either <math>r</math> or <math>s</math> is zero, the inequality is equivalent to [[Hölder's Inequality]]. | ||
| + | |||
| + | == Equivalence with the standard form == | ||
| + | |||
| + | For <math>r>s>0</math>, putting <math>x_{ij}:=a_{ij}^s</math> and <math>p:=\frac rs>1</math>, the symmetrical form given above becomes | ||
| − | + | <center><math> \sum_{j=1}^{m}\biggl(\sum_{i=1}^{n}x_{ij}^p\biggr)^{1/p} | |
| + | \geq\left(\sum_{i=1}^{n}\biggl(\sum_{j=1}^{m}x_{ij}\biggr)^p\right)^{1/p}</math>.</center> | ||
| − | <math>\ | + | Putting <math>m=2</math> and <math>a_i:=x_{i1},b_i:=x_{i2}</math>, we get the form in which the Minkowski Inequality is given most often: |
| + | |||
| + | <center><math>\biggl(\sum_{i=1}^{n}a_i^p\biggr)^{1/p}+ \biggl(\sum_{i=1}^{n}b_i^p\biggr)^{1/p} | ||
| + | \geq\left(\sum_{i=1}^{n}\Bigl(a_i+b_i\Bigr)^p\right)^{1/p}</math></center> | ||
| − | + | As the latter can be iterated, there is no loss of generality by putting <math>m=2</math>. | |
== 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: | + | [[Category:Algebra]] |
| − | [[Category: | + | [[Category:Inequalities]] |
Latest revision as of 16:49, 29 December 2021
The Minkowski Inequality states that if 
are nonzero real numbers, then for any positive numbers 
the following holds:
Notice that if either 
or 
is zero, the inequality is equivalent to Hölder's Inequality.
Equivalence with the standard form
For 
, putting 
and 
, the symmetrical form given above becomes
.Putting 
and 
, we get the form in which the Minkowski Inequality is given most often:
As the latter can be iterated, there is no loss of generality by putting .
Problems
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