Difference between revisions of "Minkowski Inequality"
Spanferkel (talk | contribs) m (→Equivalence with the standard form) |
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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 | 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} | + | <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>. | + | \geq\left(\sum_{i=1}^{n}\biggl(\sum_{j=1}^{m}x_{ij}\biggr)^p\right)^{1/p}</math>.</center> |
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: | 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} | + | <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> | + | \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> . | + | As the latter can be iterated, there is no loss of generality by putting <math>m=2</math>. |
== Problems == | == Problems == | ||
Revision as of 14:50, 11 March 2011
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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