Difference between revisions of "H�lder's inequality"
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− | + | ''See also: [[Hölder's Inequality]]'' | |
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+ | '''Hölder's Inequality,''' a generalization of the '''Cauchy-Schwarz inequality''', states that, | ||
For all <math>a_i, b_i > 0 , p,q > 0</math> such that <math>\frac {1}{p}+ \frac {1}{q} =1,</math> we have:<br> | For all <math>a_i, b_i > 0 , p,q > 0</math> such that <math>\frac {1}{p}+ \frac {1}{q} =1,</math> we have:<br> | ||
<math>\sum_{i =1}^n a_ib_i\leq \left(\sum_{i=1}^n a_i^p\right)^{\frac {1}{p}}\left(\sum _{i =1}^n b_i^q\right)^{\frac {1}{q}}.</math> | <math>\sum_{i =1}^n a_ib_i\leq \left(\sum_{i=1}^n a_i^p\right)^{\frac {1}{p}}\left(\sum _{i =1}^n b_i^q\right)^{\frac {1}{q}}.</math> |
Latest revision as of 11:12, 29 October 2016
See also: Hölder's Inequality
Hölder's Inequality, a generalization of the Cauchy-Schwarz inequality, states that,
For all such that we have:
Letting in this inequality leads to the Cauchy-Schwarz Inequality.
This can also be generalized further to sets of variables with a similar form.
Applications
1. Given we have,
2. Power-mean inequality: For and we have