Difference between revisions of "Minkowski Inequality"
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== Equivalence with the standard form == | == 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 above becomes | + | 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 |
<math> \sum_{j=1}^{m}\biggl(\sum_{i=1}^{n}x_{ij}^p\biggr)^{1/p} | <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>. | \geq\left(\sum_{i=1}^{n}\biggl(\sum_{j=1}^{m}x_{ij}\biggr)^p\right)^{1/p}</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: | |
<math>\biggl(\sum_{i=1}^{n}a_i^p\biggr)^{1/p}+ \biggl(\sum_{i=1}^{n}b_i^p\biggr)^{1/p} | <math>\biggl(\sum_{i=1}^{n}a_i^p\biggr)^{1/p}+ \biggl(\sum_{i=1}^{n}b_i^p\biggr)^{1/p} | ||
Revision as of 13:06, 12 November 2010
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.
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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