Difference between revisions of "Minkowski Inequality"
Spanferkel (talk | contribs) (→Equivalence with the standard form) |
Spanferkel (talk | contribs) (→Equivalence with the standard form) |
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<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} | ||
| − | \geq\left(\sum_{i=1}^{n}\ | + | \geq\left(\sum_{i=1}^{n}\Bigl(a_i+b_i\Bigr)^p\right)^{1/p}</math> |
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> . | ||
Revision as of 12:59, 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 above becomes
.
Put 
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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