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.

Theorem (Hölder's inequality). Let

(S, Σ, μ)

be a measure space and let

p, q \in

[1, \infty]

with

1/ p + 1/ q = 1

. Then, for all measurable real- or complex-valued functions

f

and

g

S

\|fg\|_1 \le \|f\|_p \|g\|_q.

If, in addition,

p, q \in

(1, \infty)

and

f \in L p (μ)

and

g \in L q (μ)

, then Hölder's inequality becomes an equality if and only if

| f | p

and

| g | q

are linearly dependent in

L 1 (μ)

, meaning that there exist real numbers

α, β \geq 0

, not both of them zero, such that

α | f | p = β | g | q

μ

-almost everywhere.

The numbers $p$ and $q$ above are said to be Hölder conjugates of each other. The special case $p = q = 2$ gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if $| | fg | | 1$ is infinite, the right-hand side also being infinite in that case. Conversely, if $f$ is in $L p (μ)$ and $g$ is in $L q (μ)$ , then the pointwise product $fg$ is in $L 1 (μ)$ .

Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space $L p (μ)$ , and also to establish that $L q (μ)$ is the dual space of $L p (μ)$ for $p \in$ $[1, \infty)$ .

Hölder's inequality was first found by Rogers (1888), and discovered independently by Hölder (1889).

Remarks

Conventions

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

In the definition of Hölder conjugates, $1/ \infty$ means zero.

If $p, q \in$ $[1, \infty)$ , then $| | f | | p$ and $| | g | | q$ 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 $p = \infty$ , then $| | f | | \infty$ stands for the essential supremum of $| f |$ , similarly for $| | g | | \infty$ .

The notation $| | f | | p$ with $1 \leq p \leq \infty$ is a slight abuse, because in general it is only a norm of $f$ if $| | f | | p$ is finite and $f$ is considered as equivalence class of $μ$ -almost everywhere equal functions. If $f \in L p (μ)$ and $g \in L q (μ)$ , then the notation is adequate.

On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying $a > 0$ with ∞ gives ∞.

Estimates for integrable products

As above, let $f$ and $g$ denote measurable real- or complex-valued functions defined on $S$ . If $| | fg | | 1$ is finite, then the pointwise products of $f$ with $g$ 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 $fg$ hold, and Hölder's inequality can be applied to the right-hand side. In particular, if $f$ and $g$ are in the Hilbert space $L 2 (μ)$ , then Hölder's inequality for $p = q = 2$ implies

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

where the angle brackets refer to the inner product of $L 2 (μ)$ . This is also called Cauchy–Schwarz inequality, but requires for its statement that $| | f | | 2$ and $| | g | | 2$ are finite to make sure that the inner product of $f$ and $g$ is well defined. We may recover the original inequality (for the case $p = 2$ ) by using the functions $| f |$ and $| g |$ in place of $f$ and $g$ .

Generalization for probability measures

If $(S, Σ, μ)$ is a probability space, then $p, q \in$ $[1, \infty]$ just need to satisfy $1/ p + 1/ q \leq 1$ , 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 $f$ and $g$ on $S$ ,

Notable special cases

For the following cases assume that $p$ and $q$ are in the open interval $(1,\infty)$ with $1/ p + 1/ q = 1$ .

Counting measure

For the $n$ -dimensional Euclidean space, when the set $S$ is ${1, ..., n}$ 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.

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 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 $(Ω, F, ℙ)$ , let $E$ denote the expectation operator. For real- or complex-valued random variables $X$ and $Y$ on $Ω$ , Hölder's inequality reads

\mathbb{E}[|XY|] \le \bigl(\mathbb{E}\bigl[|X|^p\bigr]\bigr)^{\frac{1}{p}} \bigl(\mathbb{E}\bigl[|Y|^q\bigr]\bigr)^{\frac{1}{q}}.

Let $0 < r < s$ and define $p = s / r$ . Then $q = p /(p - 1)$ is the Hölder conjugate of $p$ . Applying Hölder's inequality to the random variables $| X | r$ and $1 Ω$ , we obtain

\mathbb{E}\bigl[|X|^r\bigr]\le\bigl(\mathbb{E}\bigl[|X|^s\bigr]\bigr)^{r/s}.

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

Product measure

For two σ-finite measure spaces $(S 1, Σ 1, μ 1)$ and $(S 2, Σ 2, μ 2)$ 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 $S 1$ and $S 2$ , the σ-algebra $Σ$ arises as product σ-algebra of $Σ 1$ and $Σ 2$ , and $μ$ denotes the product measure of $μ 1$ and $μ 2$ . Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If $f$ and $g$ are $Σ$ -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 $(S, Σ, μ)$ denote a σ-finite measure space and suppose that $f = (f 1, ..., f n)$ and $g = (g 1, ..., g n)$ are $Σ$ -measurable functions on $S$ , taking values in the $n$ -dimensional real- or complex Euclidean space. By taking the product with the counting measure on ${1, ..., n}$ , 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 $α, β \geq 0$ , 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 $x$ in $S$ .

This finite-dimensional version generalizes to functions $f$ and $g$ taking values in a sequence space.

Proof of Hölder's inequality

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

If $| | f | | p = 0$ , then $f$ is zero $μ$ -almost everywhere, and the product $fg$ is zero $μ$ -almost everywhere, hence the left-hand side of Hölder's inequality is zero. The same is true if $| | g | | q = 0$ . Therefore, we may assume $| | f | | p > 0$ and $| | g | | q > 0$ in the following.

If $| | f | | p = \infty$ or $| | g | | q = \infty$ , then the right-hand side of Hölder's inequality is infinite. Therefore, we may assume that $| | f | | p$ and $| | g | | q$ are in $(0, \infty)$ .

If $p = \infty$ and $q = 1$ , then $| fg | \leq | | f | | \infty | g |$ almost everywhere and Hölder's inequality follows from the monotonicity of the Lebesgue integral. Similarly for $p = 1$ and $q = \infty$ . Therefore, we may also assume $p, q \in$ $(1, \infty)$ .

\|f\|_p = \|g\|_q = 1.

We now use Young's inequality, which states that

a b \le \frac{a^p}p + \frac{b^q}q

for all nonnegative $a$ and $b$ , where equality is achieved if and only if $a p = b q$ . Hence

|f(s)g(s)| \le \frac{|f(s)|^p}p + \frac{|g(s)|^q}q,\qquad s\in S.

Integrating both sides gives

\|fg\|_1 \le \frac{1}p + \frac{1}q = 1,

which proves the claim.

Under the assumptions $p \in$ $(1, \infty)$ and $| | f | | p = | | g | | q$ , equality holds if and only if $| f | p = | g | q$ almost everywhere. More generally, if $| | f | | p$ and $| | g | | q$ are in $(0, \infty)$ , then Hölder's inequality becomes an equality if and only if there exist real numbers $α, β > 0$ , namely

\alpha=\|g\|_q^q, \qquad \beta=\|f\|_p^p,

such that

\alpha |f|^p = \beta |g|^q\,

μ-almost everywhere (*).

Alternate (simpler) proof using Jensen's inequality

Extremal equality

Statement

Assume that $1 \leq p < \infty$ and let $q$ denote the Hölder conjugate. Then, for every $f \in L p (μ)$ ,

\|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 $g$ maximizing the right-hand side. When $p = \infty$ and if each set $A$ in the σ-field $Σ$ with $μ (A) = \infty$ contains a subset $B \in Σ$ with $0 < μ (B) < \infty$ (which is true in particular when $μ$ 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 $A$ in the σ-field $Σ$ with $μ (A) = \infty$ that has no subset $B \in Σ$ with $0 < μ (B) < \infty$ (the simplest example is the σ-field $Σ$ containing just the empty set and $S$ , and the measure $μ$ with $μ (S) = \infty$ . Then the indicator function $1 A$ satisfies $| | 1 A | | \infty = 1$ , but every $g \in L 1 (μ)$ has to be $μ$ -almost everywhere constant on $A$ , because it is $Σ$ -measurable, and this constant has to be zero, because $g$ is $μ$ -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 $S$ denote the natural numbers (without zero), $Σ$ the power set of $S$ , and $μ$ the counting measure. Define $f (n) = (n - 1)/ n$ for every natural number $n$ . Then $| | f | | \infty = 1$ . For $g \in L 1 (μ)$ with $0 < | | g | | 1 \leq 1$ , let $m$ denote the smallest natural number with $g (m) \neq 0$ . Then

\Bigl|\int_S fg\,\mathrm{d}\mu\Bigr| \le \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 $| | f 1 + f 2 | | p \leq | | f 1 | | p + | | f 2 | | p$ for all $f 1$ and $f 2$ in $L p (μ)$ , see Minkowski inequality.

Hölder's inequality implies that every $f \in L p (μ)$ defines a bounded (or continuous) linear functional $κ f$ on $L q (μ)$ 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

κ f

as element of the continuous dual space

L q (μ) *

coincides with the norm of

f

L p (μ)

(see also the

L p

-space article).

Generalization of Hölder's inequality

Assume that $r \in$ $(0, \infty)$ and $p 1, \dots, p n \in$ $(0, \infty]$ such that

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

Then, for all measurable real- or complex-valued functions $f 1, \dots, f n$ defined on $S$ ,

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

In particular,

f_k\in L^{p_k}(\mu)\;\;\forall k\in\{1,\ldots,n\}\implies\prod_{k=1}^n f_k \in L^r(\mu).

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

Proof of the generalization

Interpolation

Let $p 1, ..., p n \in$ $(0, \infty]$ and let $θ 1, ..., θ n \in (0, 1)$ denote weights with $θ 1 + ... + θ n = 1$ . Define $p$ as the weighted harmonic mean, i.e.,

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

Given a measurable real- or complex-valued function $f$ on $S$ , define

f_k=|f|^{\theta_k},\quad k\in\{1,\ldots,n\}.

Then by the above generalization of Hölder's inequality,

\|f\|_p=\biggl\|\prod_{k=1}^n f_k\biggr\|_p\le \prod_{k=1}^n \|f_k\|_{p_k/\theta_k}=\prod_{k=1}^n \|f\|_{p_k}^{\theta_k}.

In particular, taking $θ 1 = θ$ and $θ 2 = 1 - θ$ , in the case $n = 2$ , we obtain the interpolation result (Littlewood's inequality)

\| f\|_{p_\theta}\le \|f\|_{p_1}^\theta \cdot \|f\|_{p_0}^{1-\theta},

for $θ \in$ $(0, 1)$ and

\frac 1p_\theta= \frac \theta {p_1}+ \frac {1-\theta}{p_0}.

A similar application of Hölder gives Lyapunov's inequality: If $p=\theta p_0+(1-\theta)p_1$ for $\theta\in(0,1)$ , then

\|f\|_p^p\le\|f\|_{p_0}^{p_0\cdot\theta}\cdot\|f\|_{p_1}^{p_1(1-\theta)}.

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

Reverse Hölder inequality

Assume that $p \in (1, \infty)$ and that the measure space $(S, Σ, μ)$ satisfies $μ (S) > 0$ . Then, for all measurable real- or complex-valued functions $f$ and $g$ on $S$ such that $g (s) \neq 0$ for $μ$ -almost all $s \in S$ ,

\|fg\|_1\ge\|f\|_{\frac{1}{p}}\,\|g\|_{-1/(p-1)}.

If $| | fg | | 1 < \infty$ and $| | g | | -1/(p -1) > 0$ , then the reverse Hölder inequality is an equality if and only if there exists an $α \geq 0$ such that

|f| = \alpha|g|^{-p/(p-1)}\,

μ

-almost everywhere.

\biggl(\int_S|f|^{\frac{1}{p}}\,\mathrm{d}\mu\biggr)^{\!p}

and

\biggl(\int_S|g|^{-1/(p-1)}\,\mathrm{d}\mu\biggr)^{-(p-1)}.

Proof of the reverse Hölder inequality

Conditional Hölder inequality

Let $(Ω, F, ℙ)$ be a probability space, $G \subset F$ a sub-σ-algebra, and $p, q \in$ $(1, \infty)$ Hölder conjugates, meaning that $1/ p + 1/ q = 1$ . Then, for all real- or complex-valued random variables $X$ and $Y$ on $Ω$ ,

\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 $Z$ 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 $a > 0$ with ∞ gives ∞.

Proof of the conditional Hölder inequality

Hölder's inequality for increasing seminorms

Let $S$ be a set and let $F (S, C)$ be the space of all complex-valued functions on $S$ . Let $N$ be an increasing seminorm on $F (S, C)$ , meaning that, for all real-valued functions $f$ and $g$ in $F (S, C)$ , if $f (s) \geq g (s) \geq 0$ for all $s \in S$ , then $N (f) \geq N (g)$ . The seminorm is also allowed to attain the value ∞. Then, for all $p, q \in$ $(1, \infty)$ with $1/ p + 1/ q = 1$ , which means that they are conjugate Hölder exponents, and for all complex-valued functions $f, g \in F (S, C)$ ,^[1]

N(|fg|) \le \bigl(N(|f|^p)\bigr)^{\frac{1}{p}} \bigl(N(|g|^q)\bigr)^{\frac{1}{q}}.

Remark: If $(S, Σ, μ)$ is a measure space and $N (f)$ is the upper Lebesgue integral of the absolute value $| f |$ , then the restriction of $N$ to all $Σ$ -measurable functions gives the usual version of Hölder's inequality.

Citations

↑ For a proof see (Trèves 1967, Lemma 20.1, pp. 205–206).

References

Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1934), Inequalities, Cambridge University Press, pp. XII+314, ISBN 0-521-35880-9, JFM 60.0169.01, Zbl 0010.10703 .
Hölder, O. (1889), "Ueber einen Mittelwertsatz", Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, Band (in German) 1889 (2): 38–47, JFM 21.0260.07 . Available at Digi Zeitschriften.
Kuptsov, L. P. (2001), "Hölder inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 .
Rogers, L. J. (February 1888), "An extension of a certain theorem in inequalities", Messenger of Mathematics, New Series XVII (10): 145–150, JFM 20.0254.02, archived from the original on August 21, 2007 .
Trèves, François (1967), Topological Vector Spaces, Distributions and Kernels, Pure and Applied Mathematics. A Series of Monographs and Textbooks 25, New York, London: Academic Press, MR 0225131, Zbl 0171.10402 .

External links

Kuttler, Kenneth (2007), An Introduction to Linear Algebra (PDF), Online e-book in PDF format, Brigham Young University .
Lohwater, Arthur (1982), Introduction to Inequalities (PDF) .
Tisdell, Chris (2012), Holder's Inequality, Online video on Dr Chris Tisdell's YouTube channel .

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