INEQUALITIES: Cauchy Inequality & Chebyshev's Inequality

Cauchy's Inequality

$G = \sqrt[n]{a_1a_2\dots a_n} \leq \dfrac{a_1 + a_2 + \dots + a_n}{n} = A$

$a_1^{\lambda_1} \dot a_2^{\lambda_2} \dots a_n^{\lambda_n} \leq \lambda_1a_1 + \lambda_2a_2 + \dots + \lambda_na_n$

where 

$\lambda_1 + \lambda_2 + \dots + \lambda_n = 1$ 

and $a_i > 0, i = 1, 2, ... n$.

Chebyshev's inequality

If $a_1 \leq a_2 \leq ... \leq a_n$ and $b_1 \leq b_2 \leq ... \leq b_n$ then:

$\dfrac{a_1 + a_2 + \dots + a_n}{n} . \dfrac{b_1 + b_2 + \dots + b_n}{n} \leq $

$\leq\dfrac{a_1b_1 + a_2b_2 + \dots + a_nb_n}{n}$.

It is an equation when 

$a_1 = a_2 = ... = a_n$ and $b_1 = b_2 = ... = b_n$.