Common Estimates and Inequalities

Inequalities

What are some common inequalities?

Trivial Inequality: $x^2 \geq 0.$ Equality holds if $x=0$.

AM-GM: $\frac{a_1+a_2+...+a_n}{2} \geq \sqrt[n]{a_1a_2...a_n}$. Equality holds if and only if all$a_i$ are equal.

Cauchy-Schwarz Inequality: $(a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2) \geq (a_1b_1+a_2b_2+...+a_nb_n)^2.$ Equality holds if and only if either $a_i=0$ for all i, or there exists a constant t such that $b_i=ta_i$ for all $i$.

Mean Inequality Chain: $QM\geq AM\geq GM\geq HM$

$QM=\sqrt{\frac{a_1^2+a_2^2+...+a_n^2}{n}}$

$HM=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_n}}$

$\sqrt{\frac{a_1^2+a_2^2+...+a_n^2}{n}} \geq \frac{a_1+a_2+...+a_n}{2} \geq \sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+...+\frac{1}{a_n}}$

Rearrangement Inequality: If we have the sequences $x_1,x_2,...x_n$ and $y_1,y_2,...,y_n$, where $x_1\leq x_2\leq...\leq x_n$ and $y_1\leq y_2\leq...\leq y_n$, the largest possible "grouping" of these elements is $x_1y_1+x_2y_2+...+x_ny_n$, and the smallest possible "grouping" is $x_1y_n+x_2y_{n-1}+...+x_ny_1.$