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|<math>\vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert \leqq \left ( \vert a_1 \vert ^p + \vert a_2 \vert ^p + \cdots + \vert a_n \vert ^p \right ) ^{1/p} \left ( \vert b_1 \vert ^q + \vert b_2 \vert ^q + \cdots + \vert b_n \vert ^q \right ) ^{1/q} </math> | |<math>\vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert \leqq \left ( \vert a_1 \vert ^p + \vert a_2 \vert ^p + \cdots + \vert a_n \vert ^p \right ) ^{1/p} \left ( \vert b_1 \vert ^q + \vert b_2 \vert ^q + \cdots + \vert b_n \vert ^q \right ) ^{1/q} </math> | ||
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| + | |<math> {1 \over p} + {1 \over q} = 1 \qquad \qquad p > 1. q > 1 </math> | ||
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| + | |<math> \vert a_1 \vert ^{p-1} / \vert b_1 \vert = \vert a_2 \vert ^{p-1} / \vert b_2 \vert = \cdots =\vert a_n \vert ^{p-1} / \vert n_1 \vert . \mbox{ for } p=q=2 \mbox{ , the formula reduces to Cauchy-Shwartz Inequality} </math> | ||
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! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Tchebytchev Inequality | ! style="background: rgb(238, 238, 238) none repeat scroll 0% 0%; -moz-background-clip: -moz-initial; -moz-background-origin: -moz-initial; -moz-background-inline-policy: -moz-initial;" colspan="2" | Tchebytchev Inequality | ||
Revision as of 08:38, 25 November 2010
| Inequalities | |
|---|---|
| Triangular Inequalities | |
| $ \vert a_1 \vert - \vert a_2 \vert \leqq \vert a_1 +a_2 \vert \leqq \vert a_1 \vert + \vert a_2 \vert $ | |
| $ \vert a_1 + a_2 + \cdots + a_n \vert \leqq \vert a_1 \vert + \vert a_2 \vert + \cdots + \vert a_n \vert $ | |
| Cauchy-schwarz Inequality | |
| $ \vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert ^2 \leqq \left ( \vert a_1 \vert ^2 + \vert a_2 \vert ^2 + \cdots + \vert a_n \vert ^2 \right ) \left ( \vert b_1 \vert ^2 + \vert b_2 \vert ^2 + \cdots + \vert b_n \vert ^2 \right ) $ | |
| $ \mbox{ The equality is valid if and only if } a_1/b_1 = a_2/b_2 = \cdots = a_n/b_n $ | |
| Inequalities Involving Arithmetic, Geometric and Harmonic | |
| $ \mbox{ if } A, \ G \mbox{ and } H \mbox{ are arithmatic, geometric and harmonic means of a positive numbers } a_1 , a_2 ,\cdots , a_n, \mbox{ then } $ | |
| $ H \leqq G \leqq A $ | |
| $ A = \frac{a_1 + a_2 + \cdots + a_n}{n} \qquad \qquad G = \sqrt[n]{a_1a_2 \cdots a_n} \qquad \qquad \frac{1}{H} = \frac{1}{n} \left ( {1 \over a_1} + {1 \over a_2 }+ \cdots +{1 \over a_n } \right ) $ | |
| $ \mbox{ the equality occures only if } a_1 = a_2 =\cdots = a_n. $ | |
| Holder Inequality | |
| $ \vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert \leqq \left ( \vert a_1 \vert ^p + \vert a_2 \vert ^p + \cdots + \vert a_n \vert ^p \right ) ^{1/p} \left ( \vert b_1 \vert ^q + \vert b_2 \vert ^q + \cdots + \vert b_n \vert ^q \right ) ^{1/q} $ | |
| $ {1 \over p} + {1 \over q} = 1 \qquad \qquad p > 1. q > 1 $ | |
| $ \vert a_1 \vert ^{p-1} / \vert b_1 \vert = \vert a_2 \vert ^{p-1} / \vert b_2 \vert = \cdots =\vert a_n \vert ^{p-1} / \vert n_1 \vert . \mbox{ for } p=q=2 \mbox{ , the formula reduces to Cauchy-Shwartz Inequality} $ | |
| Tchebytchev Inequality | |
| Minkowski Inequality | |
| Cauchy-schwarz Inequality for Integrals | |
| Holder Inequality for Integrals | |
| Minkowski Inequality for Integrals | |
