Holder's inequality
If $\dfrac{1}{p} + \dfrac{1}{q} = 1, p > 1, q > 1$, then:
$|a_1b_1 + a_2b_2 + \dots + a_nb_n| \leq$
$\leq(|a_1|^p + |a_2|^p + \dots + |a_n|^p)^\dfrac{1}{p}(|b_1|^q + |b_2|^q + \dots + |b_n|^q)^\dfrac{1}{q}$
It is an equation when
$\dfrac{|a_1|^{p-1}}{|b_1|} = \dfrac{|a_2|^{p-1}}{|b_2|} = \dots = \dfrac{|a_n|^{p-1}}{|b_n|}$.
Cauchy-Schwarz (Bunyakovsky) inequality
Cauchy-Schwarz (Bunyakovsky) inequality is obtained by Holder's inequality, when $p = q = 2$:
$a_1b_1 + a_2b_2 + \dots + a_nb_n \leq$
$\leq\sqrt{(a_1^2 + a_2^2 + \dots + a_n^2)}\sqrt{(b_1^2 + b_2^2 + \dots + b_n^2)}$
It is an equality when
$\dfrac{a_1}{b_1} = \dfrac{a_2}{b_2} = \dots = \dfrac{a_n}{b_n}$.
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