Useful inequalities

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 $$a_1,...,a_n,b_1,...,b_n\in\mathbb{R}$$