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.

Formulation $2$

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

Hölder's Inequality for Sums 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 Hölder's Inequality for Sums 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

Hölder's Inequality for Sums 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

• Hölder's Inequality for Integrals
• Minkowski's Inequality for Sums

Source of Name

This entry was named for Otto Ludwig Hölder.

Historical Note

Hölder's Inequality for Sums 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