INEQUALITIES: Cauchy Inequality & Chebyshev's Inequality

Cauchy's Inequality

$G=\sqrt[n]{a_1{a}_2\dots a_n}\le {\displaystyle \frac{a_1+a_2+\cdots +a_n}{n}}=A$

$a_1^{{\lambda }_1}{\dot{a}}_2^{{\lambda }_2}\dots a_n^{{\lambda }_n}\le {\lambda }_1{a}_1+{\lambda }_2{a}_2+\cdots +{\lambda }_n{a}_n$

where 

${\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n=1$ 

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

Chebyshev's inequality

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

${\displaystyle \frac{a_1+a_2+\cdots +a_n}{n}}.{\displaystyle \frac{b_1+b_2+\cdots +b_n}{n}}\le$

$\le {\displaystyle \frac{a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n}{n}}$.

It is an equation when 

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