Difference between revisions of "Inequalities" - Rhea

Latest revision as of 14:41, 26 February 2015

keywords: triangle, Cauchy-Schwartz, Holder, Tchebytchev, Minkowski

Inequalities

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
$\vert a_1 b_1 + a_2b_2 + \cdots + a_nb_n \vert \leqq \left ( \vert a_1 \vert ^p + \vert a_2 \vert ^p + \cdots + \vert a_n \vert ^p \right ) ^{1/p} \left ( \vert b_1 \vert ^q + \vert b_2 \vert ^q + \cdots + \vert b_n \vert ^q \right ) ^{1/q}$
${1 \over p} + {1 \over q} = 1 \qquad p > 1, \ q > 1.$
$\mbox{ The equality occures only if } \vert a_1 \vert ^{p-1} / \vert b_1 \vert = \vert a_2 \vert ^{p-1} / \vert b_2 \vert = \cdots =\vert a_n \vert ^{p-1} / \vert b_n \vert .$
$\text{for} \ p = q = 2,\ \text{the formula reduces to Cauchy-Shwartz Inequality.}$
Tchebytchev Inequality
$\mbox{ if } a_1 \geqq a_2 \geqq \cdots \geqq a_n \mbox{ and } b_1 \geqq b_2 \geqq \cdots \geqq b_n \mbox{ then }$
$\left ( \frac{a_1 + a_1 + \cdots + a_n}{n} \right ) \left ( \frac{ b_1 + b_2 + \cdots +b_n}{n} \right ) \leqq \frac{a_1b_1+a_2b_2+\cdots+a_nb_n}{n}$
$(a_1 + a_2 + \cdots +a_n)(b_1 + b_2 + \cdots +b_n) \leqq n(a_1b_1 + a_2b_2 + \cdots +a_nb_n)$
Minkowski Inequality
$\mbox{ if } a_1,a_2, \cdots , a_n, b_1,b_2, \cdots, b_n \mbox{ are all positive and } p > 1 \mbox{ then }$
$\left \{ (a_1+b+1)^p + (a_2+b_2)^p+ \cdots + (a_n+b_n)^p \right \} ^{1/p} \leqq (a_1^p + a_2^p + \cdots + a_n^p)^{1/p} + (b_1^p+b_2^p+ \cdots+ b_n^p)^{1/p}$
$\mbox{ the equality holds if and only if } a_1/b_1 = a_2 /b_2 = \cdots = a_n/b_n.$
Cauchy-schwarz Inequality for Integrals
$\left \vert \int_a^b f(x) g(x) d x \right \vert ^2 \leqq \left \{ \int_a^b \vert f(x) \vert ^2 d x \right \}\left \{ \int_a^b \vert g(x) \vert ^2 d x \right \}$
$\mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad$
Holder Inequality for Integrals
$\int_a^b \vert f(x) g(x) \vert d x \leqq \left \{ \int_a^b \vert f(x) \vert ^p d x \right \} ^{1/p} \left \{ \int _a^b \vert g(x) \vert ^q d x \right \} ^{1/q}$
$\mbox { where } \frac{1}{p} + \frac{1}{q} = 1,\ p>1,\ q> 1.$
$\mbox{ if } p = q =2, \mbox{ this formula reduces to Cauchy-Schwartz inequality for intergrals } \quad$
$\mbox{ Equality holds only if } \vert f(x) \vert ^{p-1} / \vert g(x) \vert \mbox { is constant. }$
Minkowski Inequality for Integrals
$\mbox{ if } p > 1 , \quad$
$\left \{ \int_a^b \vert f(x) + g(x) \vert ^p d x \right \} ^{1/p} \leqq \left \{ \int_a^b \vert f(x) \vert ^ p d x \right \} ^{1/p} + \left \{ \int_a^b \vert g(x) \vert ^p \right \} ^{1/p}$
$\mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad$

Back to Collective Table

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+[[Category:Formulas]]
+keywords: triangle, Cauchy-Schwartz, Holder, Tchebytchev, Minkowski
+<center><font size= 4>
+'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
+</font size>
+'''Inequalities'''
+click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]]
+</center>
+----
 {|
 |-
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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> {1 \over p} + {1 \over q} = 1 \qquad \qquad p > 1. q > 1 </math>
+|<math> {1 \over p} + {1 \over q} = 1 \qquad p > 1, \ q > 1. </math>
 |-
-|<math> \mbox{ The equality occures only if } \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}</math>
+|<math> \mbox{ The equality occures only if } \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 b_n \vert . </math>
 |-
-|<math> \mbox{ reduces to Cauchy-Shwartz Inequality} </math>
+|<math> \text{for} \ p = q = 2,\ \text{the formula reduces to Cauchy-Shwartz Inequality.} </math>
 |-
 ! 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
+|-
+|<math>\mbox{ if } a_1 \geqq a_2 \geqq \cdots \geqq a_n \mbox{ and } b_1 \geqq b_2 \geqq \cdots \geqq b_n \mbox{ then } </math>
+|-
+|<math>\left ( \frac{a_1 + a_1 + \cdots + a_n}{n} \right ) \left ( \frac{ b_1 +  b_2 + \cdots +b_n}{n} \right ) \leqq \frac{a_1b_1+a_2b_2+\cdots+a_nb_n}{n}</math>
+|-
+|<math>(a_1 +  a_2 + \cdots +a_n)(b_1 +  b_2 + \cdots +b_n) \leqq n(a_1b_1 +  a_2b_2 + \cdots +a_nb_n)</math>
 |-
 ! 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" |  Minkowski Inequality
+|-
+|<math>\mbox{ if } a_1,a_2, \cdots , a_n, b_1,b_2, \cdots, b_n \mbox{ are all positive and } p > 1 \mbox{ then }</math>
+|-
+|<math> \left \{ (a_1+b+1)^p + (a_2+b_2)^p+ \cdots + (a_n+b_n)^p \right \} ^{1/p} \leqq (a_1^p + a_2^p + \cdots + a_n^p)^{1/p} + (b_1^p+b_2^p+ \cdots+ b_n^p)^{1/p}</math>
+|-
+|<math> \mbox{ the equality holds if and only if } a_1/b_1 = a_2 /b_2 = \cdots = a_n/b_n.
+</math>
 |-
 ! 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 for Integrals
+|-
+|<math> \left \vert \int_a^b f(x) g(x) d x \right \vert ^2 \leqq \left \{ \int_a^b \vert f(x) \vert ^2 d x \right \}\left \{ \int_a^b \vert g(x) \vert ^2 d x \right \}</math>
+|-
+|<math> \mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad
+</math>
 |-
 ! 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" |  Holder Inequality for Integrals
+|-
+|<math> \int_a^b \vert f(x) g(x) \vert d x \leqq \left \{ \int_a^b \vert f(x) \vert ^p d x \right \} ^{1/p} \left \{ \int _a^b \vert g(x) \vert ^q d x \right \} ^{1/q} </math>
+|-
+|<math> \mbox { where } \frac{1}{p} + \frac{1}{q} = 1,\ p>1,\ q> 1. </math>
+|-
+|<math> \mbox{ if } p = q =2, \mbox{ this formula reduces to Cauchy-Schwartz inequality for intergrals } \quad </math>
+|-
+|<math> \mbox{ Equality holds only if } \vert f(x) \vert ^{p-1} / \vert g(x) \vert \mbox { is constant. }</math>
 |-
 ! 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" | Minkowski Inequality for Integrals
+|-
+|<math> \mbox{ if } p > 1 , \quad </math>
+|-
+|<math> \left \{ \int_a^b \vert f(x) + g(x) \vert ^p d x \right \} ^{1/p} \leqq \left \{ \int_a^b \vert f(x) \vert ^ p d x \right \} ^{1/p} + \left \{ \int_a^b \vert g(x) \vert ^p \right \} ^{1/p}</math>
+|-
+|<math> \mbox{ The equality ocures only if } f(x) /g(x) \mbox { is constant} . \qquad
+</math>
 |-
 |}

Difference between revisions of "Inequalities" - Rhea

Latest revision as of 14:41, 26 February 2015

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