Lemma 4.7.1 Let $a$ and $b$ be any positive numbers, and let $p$ and $q$ be any positive numbers (necessarily greater than $1$) satisfying \begin{equation} \label{eq:lemma} \frac{1}{p} + \frac{1}{q} = 1. \end{equation} Then \[\frac{1}{p}a^p + \frac{1}{q}b^q \geq ab\] with equality if and only if $a^p = b^q.$
Theorem 4.7.2 (Hölder's Inequality) Let $X$ and $Y$ be any two random variables, and let $p$ and $q$ satisfy \ref{eq:lemma}. Then \[|EXY| \leq E|XY| \leq (E|X|^p)^{1/p}(E|Y|^q)^{1/q}.\]
Theorem 4.7.3 (Cauchy-Schwarz Inequality) For any two random variables $X$ and $Y,$ \[|EXY| \leq E|XY| \leq (E|X|^2)^{1/2}(E|Y|^2)^{1/2}.\]
Theorem 4.7.4 (Minkowski's Inequality) Let $X$ and $Y$ be any two random variables. Then for $1\leq p < \infty,$ \[[E|X+Y|^p]^{1/p} \leq [E|X|^p]^{1/p} + [E|Y|^p]^{1/p}\]