Hölder's Inequality
Definition
For two constants $p, q$ greater than $1$ satisfying $\dfrac{1}{p} + \dfrac{1}{q} = 1$ and $\mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, the following inequality holds.
$$ | \left\langle \mathbf{u}, \mathbf{v} \right\rangle | = |\mathbf{u} ^{\ast} \mathbf{v}| \le ||\mathbf{u}||_{p} ||\mathbf{v}||_{q} $$
This is called Hölder’s inequality.
Explanation
It should originally be written as Hölder’s inequality, but an alternative notation is used because of the umlaut. It is a marvelous inequality in which the $p$-norm and the $q$-norm are mixed together. Although it has no great connection in terms of usage or method of proof, when $p=q=2$ it becomes the Cauchy–Schwarz inequality.
Proof
If $\mathbf{u} = \mathbb{0}$ or $\mathbf{v} = \mathbb{0}$ it is trivial, so let us consider the case where neither is $\mathbb{0}$.
For two constants $p,q$ greater than $1$ satisfying $\dfrac{1}{p} + \dfrac{1}{q} = 1$ and two positive numbers $a,b$,
$$ ab \le { {a^{p}} \over {p} } + {{b^{q}} \over {q}} $$
Substituting $a = \dfrac{ | u_{i} | }{|| \mathbf{u}||_{p} }, b = \dfrac{ | v_{i} | }{|| \mathbf{v} || _{q} }$ into Young’s inequality gives the following inequality.
$$ {{| u_{i} v_{i} |} \over {||\mathbf{u}||_{p} ||\mathbf{v}||_{q} }} \le {{1} \over {p}} {{ |u_{i}|^{p} } \over {|| \mathbf{u}||_{p}^{p} }} + {{1} \over {q}} {{ |v_{i}|^q } \over {|| \mathbf{v}||_{q}^{q} }} $$
Taking $\displaystyle \sum_{i = 1}^{n}$ on both sides of the equation above gives the following.
$$ \begin{align*} & \sum_{i=1}^{n} \dfrac{| u_{i} v_{i} |}{||\mathbf{u}||_{p} ||\mathbf{v}||_{q} } \le & \sum_{i = 1}^{n} \left( \dfrac{1}{p} \dfrac{ |u_{i}|^{p} }{|| \mathbf{u}||_{p}^{p} } + \dfrac{1}{q} \dfrac{ |v_{i}|^q }{|| \mathbf{v}||_{q}^{q} } \right) \\ \implies && \dfrac{ \sum_{i=1}^{n} | u_{i} v_{i} |}{||\mathbf{u}||_{p} ||\mathbf{v}||_{q} } \le & \dfrac{1}{p} \dfrac{ \sum_{i=1}^{n}|u_{i}|^{p} }{|| \mathbf{u}||_{p}^{p} } + \dfrac{1}{q} \dfrac{ \sum_{i=1}^{n}|v_{i}|^q }{|| \mathbf{v}||_{q}^{q} } \\ \implies && \dfrac{ |\left\langle \mathbf{u}, \mathbf{v} \right\rangle | }{||\mathbf{u}||_{p} ||\mathbf{v}||_{q} } \le & \dfrac{1}{p} \dfrac{ || \mathbf{u}||_{p}^{p} }{|| \mathbf{u}||_{p}^{p} } + \dfrac{1}{q} \dfrac{ || \mathbf{v}||_{q}^{q} }{|| \mathbf{v}||_{q}^{q} } = \dfrac{1}{p} + \dfrac{1}{q} = 1 \end{align*} $$
The third line holds by the definition of the $p$-norm $\left( \sum_{i=1}^{n}|u_{i}|^{p} \right)^{1/p} = \left\| \mathbf{u} \right\|_{p}$. Multiplying both sides by $||\mathbf{u}||_{p} ||\mathbf{v}||_{q}$ gives the following.
$$ | \left\langle \mathbf{u}, \mathbf{v} \right\rangle | \le ||\mathbf{u}||_{p} ||\mathbf{v}||_{q} $$
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