Difference between revisions of "Inequalities" - Rhea

Revision as of 08:25, 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
Tchebytchev Inequality
Minkowski Inequality
Cauchy-schwarz Inequality for Integrals
Holder Inequality for Integrals
Minkowski Inequality for Integrals

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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" |  Cauchy-schwarz Inequality
 |-
-| <math> \vert a_1b_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 )</math>
+| <math> \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 )</math>
 |-
 |<math> \mbox{ The equality is valid if and only if } a_1/b_1 = a_2/b_2 = \cdots = a_n/b_n
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 |<math> H \leqq G \leqq A</math>
 |-
-|<math> A = \frac{a_1+a_2+\cdots + a_n}{n} \qquad \qquad G = \sqrt [n]{a_a_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 ) </math>
+|<math> 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 ) </math>
 |-
 |<math> \mbox{ the equality occures only if } a_1 = a_2 =\cdots = a_n. </math>