Minkowski Inequality: Difference between revisions
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Notice that if either <math>r</math> or <math>s</math> is zero, the inequality is equivalent to [[Holder's Inequality]]. | Notice that if either <math>r</math> or <math>s</math> is zero, the inequality is equivalent to [[Holder's Inequality]]. | ||
== 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 | |||
<math>\left(\sum_{j=1}^{m}\biggl(\sum_{i=1}^{n}a_{ij}^{r}\biggr)^{s/r}\right)^{1/s} \geq\left(\sum_{i=1}^{n}\biggl(\sum_{j=1}^{m}a_{ij}^{s}\biggr)^{r/s}\right)^{1/r}</math> | |||
becomes | |||
<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>. | |||
Put <math>m=2, a_i:=x_{i1},b_i:=x_{i2}</math> and 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} | |||
\geq\left(\sum_{i=1}^{n}\biggl(a_i+b_i\biggr)^p\right)^{1/p}</math> | |||
As the latter can be iterated, there is no loss of generality by putting <math>m=2</math> . | |||
== Problems == | == Problems == | ||
Revision as of 12:56, 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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