Difference between revisions of "Useful inequalities"

Latest revision as of 20:15, 21 April 2015

Here is a list of useful inequalities:

Inequality	Name	Conditions
$\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}$	Cauchy-Schwarz inequality	For $a_1,...,a_n,b_1,...,b_n\in\mathbb{R}$
$\left(\sum^n_{i=1}\|a_i+b_i\|^p\right)^\frac{1}{p}\le\left(\sum^n_{i=1}\|a_i\|^p\right)^\frac{1}{p}+\left(\sum^n_{i=1}\|b_i\|^p\right)^\frac{1}{p}$	Minkowski's inequality	For $p\ge 1$ and $a_1,...,a_n,b_1,...,b_n\in\mathbb{R}$

@@ Line 6: / Line 6: @@
 {| class="wikitable" border="1"
 |-
-! Name
 ! Inequality
+! Name
+! Conditions
 |-
+| <math>\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}</math>
 | [[Cauchy-Schwarz inequality]]
-| For <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math> we have <math>\sum^n_{i=1}a_ib_i\le\sqrt{\sum^n_{k=1}a_i^2}\sqrt{\sum^n_{i=1}b_i^2}</math>
+| For <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math>
+|-
+| <math>\left(\sum^n_{i=1}|a_i+b_i|^p\right)^\frac{1}{p}\le\left(\sum^n_{i=1}|a_i|^p\right)^\frac{1}{p}+\left(\sum^n_{i=1}|b_i|^p\right)^\frac{1}{p}</math>
+| [[Minkowski's inequality]]
+| For <math>p\ge 1</math> and <math>a_1,...,a_n,b_1,...,b_n\in\mathbb{R}</math>
 |}