INEQUALITIES: Holder's inequality and Cauchy-Schwarz (Bunyakovsky) inequality

Holder's inequality

If ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,p>1,q>1$, then:

$|a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n|\le$ 

$\le (|a_1{|}^p+|a_2{|}^p+\cdots +|a_n{|}^p{)}^{{\displaystyle \frac{1}{p}}}(|b_1{|}^q+|b_2{|}^q+\cdots +|b_n{|}^q{)}^{{\displaystyle \frac{1}{q}}}$

It is an equation when 

${\displaystyle \frac{|a_1{|}^{p-1}}{|b_1|}}={\displaystyle \frac{|a_2{|}^{p-1}}{|b_2|}}=\cdots ={\displaystyle \frac{|a_n{|}^{p-1}}{|b_n|}}$.

Cauchy-Schwarz (Bunyakovsky) inequality

Cauchy-Schwarz (Bunyakovsky) inequality is obtained by Holder's inequality, when $p=q=2$:

$a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n\le$ 

$\le \sqrt{(a_1^2+a_2^2+\cdots +a_n^2)}\sqrt{(b_1^2+b_2^2+\cdots +b_n^2)}$

It is an equality when 

${\displaystyle \frac{a_1}{b_1}}={\displaystyle \frac{a_2}{b_2}}=\cdots ={\displaystyle \frac{a_n}{b_n}}$.