Minkowski Inequality: Difference between revisions
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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 == | ||
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} | <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: | |||
<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}\ | \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 == | ||
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{{stub}} | {{stub}} | ||
[[Category: | [[Category:Algebra]] | ||
[[Category: | [[Category:Inequalities]] | ||
Latest revision as of 15:49, 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
This article is a stub. Help us out by expanding it.
