Hölder's Inequality for Sums

Theorem

Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:

$\dfrac 1 p + \dfrac 1 q = 1$

Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.

Formulation $1$

Let $\mathbf x$ and $\mathbf y$ denote the vectors consisting of the sequences:

$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$

$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$

where ${\ell^p}_\GF$ denotes the $p$-sequence space in $\GF$.

Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.

Then:

$\mathbf x \mathbf y \in {\ell^1}_\GF$

and:

$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$

where:

$\mathbf x \mathbf y := \sequence {x_n y_n}_{n \mathop \in \N}$

$\norm {\mathbf x \mathbf y}_1$ is the $1$-norm, also known as the taxicab norm.

Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences in $\GF$ such that $\ds \sum_{k \mathop \in \N} \size {x_k}^p$ and $\ds \sum_{k \mathop \in \N} \size {y_k}^q$ are convergent.

Then:

$\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$

Finite Form

can also be seen presented in the less general form:

$\ds \sum \limits_{k \mathop = 1}^n \size {x_k y_k} \le \paren {\sum_{k \mathop = 1}^n \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \size {y_k}^q}^{1 / q}$

where the summations are finite.

Condition for Equality

Formulation $1$: Condition for Equality

$\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$

if and only if:

$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$

for some real constant $c$.

Formulation $2$: Condition for Equality

$\ds \sum_{k \mathop \in \N} \size {x_k y_k} = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$

if and only if:

$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$

for some real constant $c$.

Parameter Inequalities

Statements of will commonly insist that $p, q > 1$.

However, we note that from Positive Real Numbers whose Reciprocals Sum to 1 we have that if:

$p, q > 0$

and:

$\dfrac 1 p + \dfrac 1 q = 1$

it follows directly that $p, q > 1$.

Also known as

is also seen referred to just as Hölder's inequality.

This allows it to be confused with Hölder's inequality for integrals, so the full form is used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

Source of Name

This entry was named for Otto Ludwig Hölder.

Historical Note

was first found by Leonard James Rogers in $1888$, and discovered independently by Otto Ludwig Hölder in $1889$.

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hölder's inequality
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hölder's inequality