Inequality Revision 1
Inequalitys with multi variables
Cauchy
\[ \sum_{i=1}^n(a_i^2)\sum_{i=1}^n(b_i^2)\geq \sum_{i=1}^n (a_i b_i)^2\]
Carlson
k = 2 is Cauchy
k is the number of lists.
\[ \sum_{i=1}^n(a_i^k)\sum_{i=1}^n(b_i^k)...\sum_{i=1}^n(z_i^k)\geq \sum_{i=1}^n (a_i b_ic_i...z_i)^k\]
Jensen
can be weighed
concave
\[ f(\frac{\sum_{i=1}^n x_i}{n}) \geq \sum_{i=1}^n\frac{f(x_i)}{n}\]
convex
\[ f(\frac{\sum_{i=1}^n x_i}{n}) \leq \sum_{i=1}^n\frac{f(x_i)}{n}\]
Holder
\(\frac{1}{p} + \frac{1}{q} = 1\), \(p = q = 2\) is Cauchy
\[ \sum_{i=1}^n(a_i^p)^q\sum_{i=1}^n(b_i^q)^p\geq \sum_{i=1}^n (a_i b_i)^{pq}\]
AM-GM
Define
\[ D(p) = \lim_{k \to p}(\frac{x_i^k}{n})^{\frac{1}{k}}\]
Then
QM AM GM HM Rule
\[ D(2) \geq D(1) \geq D(0) \geq D(-1)\]