1 Hölder Inequality
In mathematical analysis, the Hölder inequality is a fundamental inequality between integrals and an indispensable tool for the study of [[lebesgue-space]].
For measure space \((S,\Sigma,\mu)\) let \(p,q\in[1,\infty]\) with \(\frac{1}{p}+\frac{1}{q}=1\) (known as Hölder conjugates). Then, for all measurable real or complex valued functions \(f\) and \(g\) on \(S\) we have that \[\|fg\|_{1}\leq \|f\|_{p}\cdot\|g\|_{q}.\]Additionally, if \(p,q\in(1,\infty)\) and \(f\in L^p(\mu)\), \(g \in L^q(\mu)\) then we obtain equality iff \(|f|^p\) and \(|g|^q\) are linearly dependent in \(L^1(\mu)\).
By considering the \(L_{p}(\Omega, \mathcal{F}, \mathbb{P})\) space of random variables \(X:\Omega \to \mathbb{R}\) where we use the \(L_{p}\) norm \[ \lVert X \rVert _{p}:=(\mathbb{E}{\lvert X \rvert ^p})^{1/p}, \] we obtain the equivalent result detailed below.
Let \(X\) and \(Y\) be random variables, \(p,q \in (1,\infty)\) and \(\frac{1}{p}+\frac{1}{q}=1\). We have that \[|\mathbb{E}[XY]|\leq \mathbb{E}|XY|\leq (\mathbb{E}|X|^p)^{1/p}\cdot (\mathbb{E}|Y|^q)^{1/q}.\]
PROOF: First assume \(X,Y\) are non-negative (extension to negative by taking modulus). Since \(p,q>1\) we have that \(x \mapsto x^p\) is convex thus by Jensen Inequality \[ \begin{align} \int_{}^{}{XY}~d{\mathbb{P}} & = \left( \int_{}^{}{Y^q}~d{\mathbb{P}} \right)\int_{}^{}{XY^{1-q}\cdot \frac{Y^q}{\int_{}^{}{Y^q}~d{\mathbb{P}}}}~d{\mathbb{P}} \\ & \leq \left( \int_{}^{}{Y^q}~d{\mathbb{P}} \right)\left( \int_{}^{}{X^pY^{p(1-q)}\cdot \frac{Y^q}{\int_{}^{}{Y^q}~d{\mathbb{P}}}}~d{\mathbb{P}} \right)^{\frac{1}{p}} \\ & =\left( \int_{}^{}{Y^q}~d{\mathbb{P}} \right)\left( \int_{}^{}{\frac{X^p}{\int_{}^{}{Y^q}~d{\mathbb{P}}}}~d{\mathbb{P}} \right) ^{\frac{1}{p}}. \end{align} \] Thus we obtain \[ \int_{}^{}{XY}~d{\mathbb{P}}\leq \left( \int_{}^{}{X^p}~d{\mathbb{P}} \right)^{\frac{1}{p}}\left( \int_{}^{}{Y^q}~d{\mathbb{P}} \right)^{\frac{1}{q}}. \] \(\square\)