Difference between revisions of "Minkowski Inequality"
Spanferkel (talk | contribs) m (→Equivalence with the standard form) |
Etmetalakret (talk | contribs) |
||
(2 intermediate revisions by one other user not shown) | |||
Line 1: | Line 1: | ||
− | 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 == | ||
Line 8: | Line 8: | ||
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 == | ||
Line 23: | Line 23: | ||
{{stub}} | {{stub}} | ||
− | [[Category: | + | [[Category:Algebra]] |
− | [[Category: | + | [[Category:Inequalities]] |
Latest revision as of 16:49, 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
![$\sum_{j=1}^{m}\biggl(\sum_{i=1}^{n}x_{ij}^p\biggr)^{1/p} \geq\left(\sum_{i=1}^{n}\biggl(\sum_{j=1}^{m}x_{ij}\biggr)^p\right)^{1/p}$](http://latex.artofproblemsolving.com/a/d/d/adddab5aa0b607e4222a3d6af63ef9c27b05fab9.png)
Putting and
, we get the form in which the Minkowski Inequality is given most often:
![$\biggl(\sum_{i=1}^{n}a_i^p\biggr)^{1/p}+ \biggl(\sum_{i=1}^{n}b_i^p\biggr)^{1/p} \geq\left(\sum_{i=1}^{n}\Bigl(a_i+b_i\Bigr)^p\right)^{1/p}$](http://latex.artofproblemsolving.com/4/5/f/45f54de6da54fe8ef65efd057f5f6b0daaff6d7b.png)
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.