Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of $L^{p}$ spaces.

The numbers $[1, ∞]$ and $f$ above are said to be Hölder conjugates of each other. The special case $(S, Σ, μ)$ gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if $p, q ∈$ is infinite, the right-hand side also being infinite in that case. Conversely, if $g$ is in $1/p + 1/q = 1$ and $S$ is in $p, q ∈$, then the pointwise product $f ∈ L^{p}(μ)$ is in $g ∈ L^{q}(μ)$.

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space $|f |^{p}$, and also to establish that $|g|^{q}$ is the dual space of $L^{1}(μ)$ for $α, β ≥ 0$ $(1, ∞)$.

Hölder's inequality (in a slightly different form) was first found by. Inspired by Rogers' work, gave another proof as part of a work developing the concept of convex and concave functions and introducing Jensen's inequality, which was in turn named for work of Johan Jensen building on Hölder's work.

Conventions
The brief statement of Hölder's inequality uses some conventions.


 * In the definition of Hölder conjugates, $α|f |^{p} = β |g|^{q}$ means zero.
 * If $p = q = 2$ $μ$, then $\|fg\|_{1}$ and $L^{p}(μ)$ stand for the (possibly infinite) expressions
 * $$\begin{align}

&\left(\int_S |f|^p\,\mathrm{d}\mu\right)^{\frac{1}{p}} \\ &\left(\int_S |g|^q\,\mathrm{d}\mu\right)^{\frac{1}{q}} \end{align}$$


 * If $L^{q}(μ)$, then $fg$ stands for the essential supremum of $L^{1}(μ)$, similarly for $L^{p}(μ)$.
 * The notation $L^{q}(μ)$ with $L^{p}(μ)$ is a slight abuse, because in general it is only a norm of $p$ if $p ∈$ is finite and $q$ is considered as equivalence class of $f$-almost everywhere equal functions. If $1/∞$ and $p, q ∈$, then the notation is adequate.
 * On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying $\|f \|_{p}$ with ∞ gives ∞.

Estimates for integrable products
As above, let $g$ and $[1, ∞)$ denote measurable real- or complex-valued functions defined on $[1, ∞)$. If $\|g\|_{q}$ is finite, then the pointwise products of $f$ with $f$ and its complex conjugate function are $μ$-integrable, the estimate


 * $$\biggl|\int_S f\bar g\,\mathrm{d}\mu\biggr|\le\int_S|fg|\,\mathrm{d}\mu =\|fg\|_1$$

and the similar one for $p = ∞$ hold, and Hölder's inequality can be applied to the right-hand side. In particular, if $f$ and $g$ are in the Hilbert space $\|f \|_{∞}$, then Hölder's inequality for $|f |$ implies


 * $$|\langle f,g\rangle| \le \|f\|_2 \|g\|_2,$$

where the angle brackets refer to the inner product of $\|g\|_{∞}$. This is also called Cauchy–Schwarz inequality, but requires for its statement that $\|f \|_{p}$ and $1 ≤ p ≤ ∞$ are finite to make sure that the inner product of $S$ and $f$ is well defined. We may recover the original inequality (for the case $\|f \|_{p}$) by using the functions $f ∈ L^{p}(μ)$ and $g ∈ L^{q}(μ)$ in place of $g$ and $μ$.

Generalization for probability measures
If $a > 0$ is a probability space, then $\|fg\|_{1}$ $f$ just need to satisfy $fg$, rather than being Hölder conjugates. A combination of Hölder's inequality and Jensen's inequality implies that


 * $$\|fg\|_1 \le \|f\|_p \|g\|_q$$

for all measurable real- or complex-valued functions $g$ and $f$ on $g$.

Notable special cases
For the following cases assume that $f$ and $g$ are in the open interval $[1, ∞]$ with $L^{2}(μ)$.

Counting measure
For the $$n$$-dimensional Euclidean space, when the set $$S$$ is $$\{1,\dots,n\}$$ with the counting measure, we have


 * $$\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\frac{1}{p}} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\frac{1}{q}}

\text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.$$ Often the following practical form of this is used, for any $$(r,s)\in\mathbb{R}_+$$:
 * $$\biggl(\sum_{k=1}^n |x_k|^r\,|y_k|^s \biggr)^{r+s}\le \biggl( \sum_{k=1}^n |x_k|^{r+s} \biggr)^{r} \biggl( \sum_{k=1}^n |y_k|^{r+s} \biggr)^{s}.$$

For more than two sums, the following generalisation holds, with real positive exponents   $$ \lambda_i $$ and $$ \lambda_a + \lambda_b+ \cdots +\lambda_z =1$$:
 * $$\sum_{k=1}^n |a_k|^{\lambda_a}\,|b_k|^{\lambda_b} \cdots |z_k|^{\lambda_z} \le \biggl(\sum_{k=1}^n |a_k|\biggr)^{\lambda_a} \biggl(\sum_{k=1}^n |b_k|\biggr)^{\lambda_b} \cdots \biggl(\sum_{k=1}^n |z_k|\biggr)^{\lambda_z}.

$$ Equality holds iff $$ |a_1|: |a_2|: \cdots : |a_n| =|b_1|: |b_2|: \cdots : |b_n| = \cdots = |z_1|: |z_2|: \cdots : |z_n| $$.

If $$S=\N$$ with the counting measure, then we get Hölder's inequality for sequence spaces:


 * $$\sum_{k=1}^{\infty} |x_k\,y_k| \le \biggl( \sum_{k=1}^{\infty} |x_k|^p \biggr)^{\frac{1}{p}} \left( \sum_{k=1}^{\infty} |y_k|^q \right)^{\frac{1}{q}}

\text{ for all }(x_k)_{k\in\mathbb N}, (y_k)_{k\in\mathbb N}\in\mathbb{R}^{\mathbb N}\text{ or }\mathbb{C}^{\mathbb N}.$$

Lebesgue measure
If $$S$$ is a measurable subset of $$\R^n$$ with the Lebesgue measure, and $$f$$ and $$g$$ are measurable real- or complex-valued functions on $$S$$, then Hölder's inequality is


 * $$\int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x\biggr)^{\frac{1}{p}} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\frac{1}{q}}.$$

Probability measure
For the probability space $$(\Omega, \mathcal{F}, \mathbb{P}),$$ let $$\mathbb{E}$$ denote the expectation operator. For real- or complex-valued random variables $$X$$ and $$Y$$ on $$\Omega,$$ Hölder's inequality reads


 * $$\mathbb{E}[|XY|] \leqslant \left (\mathbb{E}\bigl[ |X|^p\bigr]\right)^{\frac{1}{p}} \left(\mathbb{E}\bigl[|Y|^q\bigr]\right)^{\frac{1}{q}}.$$

Let $$1 < r < s < \infty$$ and define $$p = \tfrac{s}{r}.$$ Then $$q = \tfrac{p}{p-1}$$ is the Hölder conjugate of $$p.$$ Applying Hölder's inequality to the random variables $$|X|^r$$ and $$1_{\Omega}$$ we obtain


 * $$\mathbb{E}\bigl[|X|^r\bigr]\leqslant \left(\mathbb{E}\bigl[|X|^s\bigr]\right)^{\frac{r}{s}}.$$

In particular, if the $f$th absolute moment is finite, then the $g$th absolute moment is finite, too. (This also follows from Jensen's inequality.)

Product measure
For two σ-finite measure spaces $p = q = 2$ and $L^{2}(μ)$ define the product measure space by


 * $$S=S_1\times S_2,\quad \Sigma=\Sigma_1\otimes\Sigma_2,\quad \mu=\mu_1\otimes\mu_2,$$

where $S$ is the Cartesian product of $\|f \|_{2}$ and $\|g\|_{2}$, the σ-algebra $p = 2$ arises as product σ-algebra of $|f |$ and $|g|$, and $p$ denotes the product measure of $(S, Σ, μ)$ and $p, q ∈$. Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If $q$ and $(1,∞)$ are $1/p + 1/q ≤ 1$-measurable real- or complex-valued functions on the Cartesian product $s$, then


 * $$\int_{S_1}\int_{S_2}|f(x,y)\,g(x,y)|\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x) \le\left(\int_{S_1}\int_{S_2}|f(x,y)|^p\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_{S_1}\int_{S_2}|g(x,y)|^q\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\right)^{\frac{1}{q}}.$$

This can be generalized to more than two σ-finite measure spaces.

Vector-valued functions
Let $1/p + 1/q = 1$ denote a σ-finite measure space and suppose that $(S_{1}, Σ_{1}, μ_{1})$ and $(S_{2}, Σ_{2}, μ_{2})$ are $S_{1}$-measurable functions on $r$, taking values in the $S$-dimensional real- or complex Euclidean space. By taking the product with the counting measure on $S_{2}$, we can rewrite the above product measure version of Hölder's inequality in the form


 * $$ \int_S \sum_{k=1}^n|f_k(x)\,g_k(x)|\,\mu(\mathrm{d}x) \le \left(\int_S\sum_{k=1}^n|f_k(x)|^p\,\mu(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_S\sum_{k=1}^n|g_k(x)|^q\,\mu(\mathrm{d}x)\right)^{\frac{1}{q}}.$$

If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbers $Σ$, not both of them zero, such that


 * $$\alpha \left (|f_1(x)|^p,\ldots,|f_n(x)|^p \right )= \beta \left (|g_1(x)|^q,\ldots,|g_n(x)|^q \right ),$$

for $μ$-almost all $f$ in $g$.

This finite-dimensional version generalizes to functions $S$ and $S$ taking values in a normed space which could be for example a sequence space or an inner product space.

Proof of Hölder's inequality
There are several proofs of Hölder's inequality; the main idea in the following is Young's inequality for products.

$n$

Alternative proof using Jensen's inequality:

$μ$

We could also bypass use of both Young's and Jensen's inequalities. The proof below also explains why and where the Hölder exponent comes in naturally.

$x$

Statement
Assume that $Σ_{1}$ and let $S$ denote the Hölder conjugate. Then for every $Σ_{2}$,


 * $$\|f\|_p = \max \left \{ \left| \int_S f g \, \mathrm{d}\mu \right | : g\in L^q(\mu), \|g\|_q \le 1 \right\},$$

where max indicates that there actually is a $f$ maximizing the right-hand side. When $μ_{1}$ and if each set $g$ in the σ-field $μ_{2}$ with $Σ$ contains a subset $(S, Σ, μ)$ with $f = (f_{1}, ..., f_{n})$ (which is true in particular when $$ is σ-finite), then


 * $$\|f\|_\infty = \sup \left\{ \left| \int_S f g \,\mathrm{d}\mu \right| : g\in L^1(\mu), \|g\|_1 \le 1 \right \}.$$

Proof of the extremal equality:

$$

Remarks and examples

 * The equality for $$p = \infty$$ fails whenever there exists a set $$A$$ of infinite measure in the $$\sigma$$-field $$\Sigma$$ with that has no subset $$B \in \Sigma$$ that satisfies: $$0 < \mu(B) < \infty.$$ (the simplest example is the $$\sigma$$-field $$\Sigma$$ containing just the empty set and $$S,$$ and the measure $$\mu$$ with $$\mu(S) = \infty.$$) Then the indicator function $$1_A$$ satisfies $$\|1_A\|_{\infty} = 1,$$ but every $$g \in L^1 (\mu)$$ has to be $$\mu$$-almost everywhere constant on $$A,$$ because it is $$\Sigma$$-measurable, and this constant has to be zero, because $$g$$ is $$\mu$$-integrable. Therefore, the above supremum for the indicator function $$1_A$$ is zero and the extremal equality fails.
 * For $$p = \infty,$$ the supremum is in general not attained. As an example, let $$S = \mathbb{N}, \Sigma = \mathcal{P}(\mathbb{N})$$ and $$\mu$$ the counting measure. Define:


 * $$\begin{cases} f: \mathbb{N} \to \mathbb{R} \\ f(n) = \frac{n-1}{n} \end{cases}$$


 * Then $$\|f\|_{\infty} = 1.$$ For $$g \in L^1 (\mu, \mathbb{N})$$ with $$0 < \|g\|_1 \leqslant 1,$$ let $$m$$ denote the smallest natural number with $$g(m) \neq 0.$$ Then


 * $$\left |\int_S fg\,\mathrm{d}\mu\right| \leqslant \frac{m-1}{m}|g(m)|+\sum_{n=m+1}^\infty|g(n)| = \|g\|_1-\frac{|g(m)|}m<1.$$

Applications

 * The extremal equality is one of the ways for proving the triangle inequality $g = (g_{1}, ..., g_{n})$ for all $Σ$ and $\{1, ..., n\}$ in $α, β ≥ 0$, see Minkowski inequality.
 * Hölder's inequality implies that every $\|f \|_{p} = 0$ defines a bounded (or continuous) linear functional $fg$ on $\|g\|_{q} = 0$ by the formula
 * $$\kappa_f(g) = \int_S f g \, \mathrm{d}\mu,\qquad g\in L^q(\mu).$$
 * The extremal equality (when true) shows that the norm of this functional $\|f \|_{p} > 0$ as element of the continuous dual space $\|g\|_{q} > 0$ coincides with the norm of $$ in $\|f \|_{p} = ∞$ (see also the $\|g\|_{q} = ∞$-space article).

Statement
Assume that $\|f \|_{p}$ $q$ and $\|g\|_{q}$ $g$ such that


 * $$\sum_{k=1}^n \frac1{p_k} = \frac1r$$

where 1/∞ is interpreted as 0 in this equation. Then for all measurable real or complex-valued functions $p = ∞$ defined on $A$,


 * $$\left\|\prod_{k=1}^n f_k\right\|_r \le \prod_{k=1}^n \left\|f_k\right\|_{p_k}$$

where we interpret any product with a factor of ∞ as ∞ if all factors are positive, but the product is 0 if any factor is 0.

In particular, if $$f_k \in L^{p_k}(\mu)$$ for all $$k \in \{ 1, \ldots, n \}$$ then $$\prod_{k=1}^n f_k \in L^r(\mu).$$

Note: For $$r \in (0, 1),$$ contrary to the notation, $q = 1$ is in general not a norm because it doesn't satisfy the triangle inequality.

Proof of the generalization: $μ$

Interpolation
Let $|fg| ≤ \|f \|_{∞} |g|$ $$ and let $p = 1$ denote weights with $q = ∞$. Define $$p$$ as the weighted harmonic mean, that is,


 * $$ \frac1p = \sum_{k=1}^n \frac{\theta_k}{p_k}.$$

Given measurable real- or complex-valued functions $$f_k$$ on $f$, then the above generalization of Hölder's inequality gives
 * $$\left\| |f_1|^{\theta_1}\cdots |f_n|^{\theta_n}\right\|_p \le \left\||f_1|^{\theta_1}\right\|_{\frac{p_1}{\theta_1}}\cdots \left\| |f_n|^{\theta_n}\right\|_{\frac{p_n}{\theta_n}} = \|f_1\|_{p_1}^{\theta_1}\cdots \|f_n\|_{p_n}^{\theta_n}.$$

In particular, taking $$f_1 = \cdots = f_n=:f$$ gives
 * $$\|f\|_p \leqslant \prod_{k=1}^n \|f\|_{p_k}^{\theta_k}.$$

Specifying further $p, q ∈$ and $\|f \|_{p}$, in the case $$n = 2,$$ we obtain the interpolation result

An application of Hölder gives

Both Littlewood and Lyapunov imply that if $$f \in L^{p_0}\cap L^{p_1}$$ then $$f \in L^p$$ for all $$p_0 < p < p_1.$$

Two functions
Assume that $\|g\|_{q}$ and that the measure space $(1,∞)$ satisfies $a^{p} = b^{q}$. Then for all measurable real- or complex-valued functions $(0, ∞]$ and $(0, ∞]$ on $S$ such that $p ∈$ for $$-almost all $\|f \|_{p} = \|g\|_{q}$,


 * $$\|fg\|_1\geqslant \|f\|_{\frac{1}{p}}\,\|g\|_{\frac{-1}{p-1}}.$$

If


 * $$\|fg\|_1 < \infty \quad \text{and} \quad \|g\|_{\frac{-1}{p-1}} > 0, $$

then the reverse Hölder inequality is an equality if and only if


 * $$\exists \alpha \geqslant 0 \quad |f| = \alpha|g|^{\frac{-p}{p-1}} \qquad \mu\text{-almost everywhere}.$$

Note: The expressions:

$$ \|f\|_{\frac{1}{p}}$$ and $$\|g\|_{\frac{-1}{p-1}},$$

are not norms, they are just compact notations for


 * $$\left (\int_S|f|^{\frac{1}{p}}\,\mathrm{d}\mu\right)^{p} \quad \text{and} \quad \left (\int_S|g|^{\frac{-1}{p-1}}\,\mathrm{d}\mu\right)^{-(p-1)}.$$

Multiple functions
The Reverse Hölder inequality (above) can be generalized to the case of multiple functions if all but one conjugate is negative. That is,


 * Let $$p_1,..., p_{m-1} < 0$$ and $$p_m \in \mathbb{R}$$ be such that $$\sum_{k=1}^{m} \frac{1}{p_k} = 1$$ (hence $$0 < p_m < 1$$). Let $$f_k$$ be measurable functions for $$k = 1,...,m$$. Then


 * $$\left\|\prod_{k=1}^m f_k\right\|_1 \ge \prod_{k=1}^m \left\|f_k\right\|_{p_k}.$$

This follows from the symmetric form of the Hölder inequality (see below).

Symmetric forms of Hölder inequality
It was observed by Aczél and Beckenbach that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function):

Let $$f = (f(1),\dots,f(m)), g = (g(1),\dots, g(m)), h = (h(1),\dots,h(m))$$ be vectors with positive entries and such that $$f(i) g(i) h(i) = 1$$ for all $$i$$. If $$p,q,r$$ are nonzero real numbers such that $$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0$$, then:
 * $$\|f\|_p \|g\|_q \|h\|_r \ge 1$$ if all but one of $$p,q,r$$ are positive;
 * $$\|f\|_p \|g\|_q \|h\|_r \le 1$$ if all but one of $$p,q,r$$ are negative.

The standard Hölder inequality follows immediately from this symmetric form (and in fact is easily seen to be equivalent to it). The symmetric statement also implies the reverse Hölder inequality (see above).

The result can be extended to multiple vectors:

Let $$f_1, \dots, f_n$$ be $$n$$ vectors in $$\mathbb{R}^m$$ with positive entries and such that $$f_1(i) \dots f_n(i) = 1$$ for all $$i$$. If $$p_1,\dots,p_n$$ are nonzero real numbers such that $$\frac{1}{p_1}+\dots+\frac{1}{p_n}=0$$, then:
 * $$\|f_1\|_{p_1} \dots \|f_n\|_{p_n} \ge 1$$ if all but one of the numbers $$p_i$$ are positive;
 * $$\|f_1\|_{p_1} \dots \|f_n\|_{p_n} \le 1$$ if all but one of the numbers $$p_i$$ are negative.

As in the standard Hölder inequalities, there are corresponding statements for infinite sums and integrals.

Conditional Hölder inequality
Let $|f |^{p} = |g|^{q}$ be a probability space, $\|f \|_{p}$ a sub-σ-algebra, and $\|g\|_{q}$ $(0, ∞]$ Hölder conjugates, meaning that $α, β > 0$. Then for all real- or complex-valued random variables $S$ and $f$ on $\|f \|_{p} = 0$,


 * $$\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr] \le \bigl(\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{p}} \,\bigl(\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{q}}

\qquad\mathbb{P}\text{-almost surely.}$$

Remarks:
 * If a non-negative random variable $g$ has infinite expected value, then its conditional expectation is defined by
 * $$\mathbb{E}[Z|\mathcal{G}] = \sup_{n\in\mathbb{N}}\,\mathbb{E}[\min\{Z,n\}|\mathcal{G}]\quad\text{a.s.}$$


 * On the right-hand side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying $β = 0$ with ∞ gives ∞.

Proof of the conditional Hölder inequality: $S$

Hölder's inequality for increasing seminorms
Let $μ$ be a set and let $$F(S, \mathbb{C})$$ be the space of all complex-valued functions on $p$. Let $p$ be an increasing seminorm on $$F(S, \mathbb{C}),$$ meaning that, for all real-valued functions $$f, g \in F(S, \mathbb{C})$$ we have the following implication (the seminorm is also allowed to attain the value ∞):


 * $$ \forall s \in S \quad f(s) \geqslant g(s) \geqslant 0  \qquad \Rightarrow \qquad  N(f) \geqslant N(g).$$

Then:


 * $$\forall f, g \in F(S, \mathbb{C}) \qquad N(|fg|) \leqslant \bigl(N(|f|^p)\bigr)^{\frac{1}{p}} \bigl(N(|g|^q)\bigr)^{\frac{1}{q}},$$

where the numbers $$p$$ and $$q$$ are Hölder conjugates.

Remark: If $\|g\|_{q} = 0$ is a measure space and $$N(f)$$ is the upper Lebesgue integral of $$|f|$$ then the restriction of $g$ to all $α = 0$-measurable functions gives the usual version of Hölder's inequality.

Distances based on Hölder inequality
Hölder inequality can be used to define statistical dissimilarity measures between probability distributions. Those Hölder divergences are projective: They do not depend on the normalization factor of densities.