Difference between revisions of "Minkowski Inequality"
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| − | The '''Minkowski Inequality''' states that if <math>r>s</math> | + | 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> | <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 [[ | + | 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 == | == Equivalence with the standard form == | ||
Revision as of 13:57, 11 March 2011
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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