Difference between revisions of "H�lder's inequality"
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<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> | ||
<br> | <br> | ||
− | Letting <math>p=q=2</math> in this inequality leads to the Cauchy-Schwarz Inequality. | + | Letting <math>p=q=2</math> in this inequality leads to the [[Cauchy-Schwarz]] Inequality. |
<br> | <br> | ||
This can also be generalized further to <math> n</math> sets of variables with a similar form. | This can also be generalized further to <math> n</math> sets of variables with a similar form. |
Revision as of 16:17, 8 February 2013
The 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