# Hölder's inequality

**Hölder's inequality** is a bound in analysis stating that the sum, or integral, of a product of functions or sequences is controlled by the product of their norms in conjugate exponents: for \( p > 1 \) and \( q \) with \( 1/p + 1/q = 1 \), \( \int |fg| \, d\mu \le \|f\|_p \, \|g\|_q \). It generalizes the [Cauchy–Schwarz inequality](https://www.edgepedia.org/cauchy-schwarz-inequality), which is the case \( p = q = 2 \), and it is the workhorse behind the duality of \( L^p \) spaces and a long list of estimates in functional analysis and partial differential equations.<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup><sup> • </sup><sup>[2](https://math.mit.edu/~setia/6710fa19.pdf)</sup>

| Key fact | Detail |
|---|---|
| Statement (integrals) | For \( 1 < p < \infty \) and \( 1/p + 1/q = 1 \), \( \int \|fg\| \, d\mu \le \|f\|_p \, \|g\|_q \)<sup>[2](https://math.mit.edu/~setia/6710fa19.pdf)</sup> |
| Statement (sums) | \( \Bigl\|\sum_{s \in S} a_s b_s\Bigr\| \le \bigl(\sum_{s \in S} \|a_s\|^p\bigr)^{1/p} \bigl(\sum_{s \in S} \|b_s\|^q\bigr)^{1/q} \) for complex numbers indexed by any set \( S \)<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> |
| Special case | \( p = q = 2 \) gives the Bunyakovskii inequality, better known in English-language literature as the Cauchy–Schwarz inequality<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> |
| Equality condition | Equality holds if and only if \( \|a_s\|^p = C\|b_s\|^q \) and \( \arg(a_s b_s) \) and \( C \) are independent of \( s \in S \)<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> |
| Limit case | \( p = 1 \), \( q = \infty \): \( \Bigl\|\sum a_s b_s\Bigr\| \le \bigl(\sum \|a_s\|\bigr) \sup_{s} \|b_s\| \); for \( 0 < p < 1 \) the inequality is reversed<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> |
| Converse | The converse for sums is also true, due to M. Riesz: if \( \Bigl\|\sum a_s b_s\Bigr\| \le AB \) for all \( \{a_s\} \) with \( \sum \|a_s\|^p \le A^p \), then \( \sum \|b_s\|^q \le B^q \)<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> |
| Origin | Otto Hölder, "Ueber einen Mittelwerthssatz", *Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen* (1889), pp. 38–47<sup>[3](https://de.wikisource.org/wiki/Ueber_einen_Mittelwerthssatz)</sup> |

## Statement and forms

The two exponents \( p \) and \( q \) satisfying \( 1/p + 1/q = 1 \) are called conjugate exponents (or Hölder conjugates). For measurable functions on a measure space \( (\Omega, \mu) \), with the \( L^p \) norm defined by \( \|f\|_p = \bigl(\int \|f\|^p \, d\mu\bigr)^{1/p} \), the inequality reads

\[ \int |fg| \, d\mu \le \|f\|_p \, \|g\|_q, \qquad 1 < p < \infty, \quad \frac{1}{p} + \frac{1}{q} = 1. \]

The Encyclopedia of Mathematics states the same result for sums of complex numbers \( \{a_s\} \) and \( \{b_s\} \) over any finite or infinite index set \( S \), and for integrals over a Lebesgue-measurable set in \( \mathbb{R}^n \); in full generality \( S \) may be any set carrying an additive function \( \mu \) (for example a measure) on an algebra of its subsets, with the functions \( \mu \)-measurable and integrable to the required degrees.<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup>

**Limit and reversed cases.** When \( p = 1 \) and \( q = \infty \), the inequality becomes \( \Bigl\|\sum_{s} a_s b_s\Bigr\| \le \bigl(\sum_{s} \|a_s\|\bigr) \sup_{s} \|b_s\| \). For \( 0 < p < 1 \) the inequality reverses direction.<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup>

## Generalized form

For \( m \) factors the inequality extends with weights \( \rho_s \ge 0 \):

\[ \Biggl\| \sum_{s \in S} \rho_s \, a_{1s} \cdot \ldots \cdot a_{ms} \Biggr\| \le \prod_{k=1}^{m} \Bigl( \sum_{s \in S} \rho_s \, \|a_{ks}\|^{p_k} \Bigr)^{1/p_k}, \qquad \frac{1}{p_1} + \ldots + \frac{1}{p_m} = 1, \quad p_k > 1. \]

The same product structure holds for integrals of \( m \) functions in \( L_{p_k}(S) \). There is also an abstract generalized Hölder inequality for any positive functional \( \phi \) that is homogeneous, monotone, subadditive, and vanishes at zero: \( \phi(\|a_1 \cdot \ldots \cdot a_m\|) \le \prod_{k=1}^{m} [\phi(\|a_k\|^{p_k})]^{1/p_k} \).<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup>

## Proof idea

The standard proof reduces by scaling to the case \( \|f\|_p = \|g\|_q = 1 \) and then applies [Young's inequality](https://www.edgepedia.org/youngs-inequality), \( uv \le u^p/p + v^q/q \), which itself follows from the concavity of the logarithm or from convexity.<sup>[2](https://math.mit.edu/~setia/6710fa19.pdf)</sup> An alternative proof uses [Jensen's inequality](https://www.edgepedia.org/jensens-inequality): defining a finite measure \( d\nu = g^{q_0} \, d\mu \) (where \( q_0 \) denotes the conjugate of \( p \)) and applying Jensen with \( J(t) = \|t\|^p \) yields the result directly.<sup>[4](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursLp.pdf)</sup>

## Role in \( L^p \) duality

Hölder's inequality is the mechanism by which functions in the conjugate space act as linear functionals. For \( 1 \le p \le \infty \) and \( p_0 \) conjugate to \( p \), each \( g \in L^{p_0} \) defines \( \Phi_g(f) = \int f g \, d\mu \), a continuous linear functional on \( L^p \) with \( \|\Phi_g\| = \|g\|_{p_0} \): Hölder gives the bound \( \|\Phi_g\| \le \|g\|_{p_0} \), and the equality case of Hölder gives the reverse. The converse statement, that every bounded linear functional on \( L^p \) for \( 1 \le p < \infty \) arises uniquely in this way, completes the duality \( (L^p)' = L^{p_0} \).<sup>[4](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursLp.pdf)</sup>

## Applications

The inequality appears throughout analysis as the step that splits a product into separately controlled factors.

**Sobolev inequalities.** In the proof of the [Sobolev inequality](https://www.edgepedia.org/sobolev-inequality) for \( W^{1,p}_0(U) \) with \( 1 \le p < d \), Hölder's inequality bounds an intermediate term by \( \gamma \, \bigl\| \|u\|^{\gamma-1} \bigr\|_{L^{p/(p-1)}} \, \|Du\|_{L^p} \), and the exponents then combine, by homogeneity, into the Sobolev exponent \( p^* \).<sup>[5](https://web.math.berkeley.edu/~sjoh/pdfs/notes-math222a.pdf)</sup> In the study of nonlinear partial differential equations, Hölder's inequality yields product estimates such as \( \|\partial^{\alpha} f \cdot \partial^{\beta} g\|_{L^2} \le C \, \|\partial^{\alpha} f\|_{L^{2k/(k-r)}} \, \|\partial^{\beta} g\|_{L^{2k/r}} \), which feed into Moser's interpolation inequality.<sup>[6](https://www.math.cuhk.edu.hk/course_builder/2526/math5021/1-1%20Sobolev%20spaces.pdf)</sup>

**Norm inclusions.** On a measure space of finite measure, Hölder's inequality shows that \( L^{p_2} \) embeds in \( L^{p_1} \) whenever \( p_1 < p_2 \), by applying the inequality to \( f^{p_2} \) against the constant function 1.<sup>[4](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursLp.pdf)</sup><sup> • </sup><sup>[11](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/sauter/ws14/measure-theory.part3.2015-02-05.pdf)</sup>

**Non-commutative settings.** Hölder-type inequalities have been extended beyond commutative \( L^p \) spaces: a Hölder inequality for KMS states in quantum statistical mechanics generalizes a well-known trace inequality and is formulated in the language of non-commutative \( L_p \)-spaces.<sup>[7](https://arxiv.org/abs/1103.3608)</sup>

## Stability: how sharp is the bound?

Equality in Hölder's inequality occurs only under a proportionality condition, and a line of work quantifies how far the two sides are apart when equality nearly holds.

J. M. Aldaz, a researcher in mathematical analysis, proved a stability version of the inequality that incorporates an extra term measuring the deviation from equality, with applications including interpolation-type estimates for \( L^p \) norms and a proof of the uniform convexity of \( L^p \) spaces.<sup>[8](https://arxiv.org/abs/0710.2307)</sup> In geometric form, Aldaz's result says that \( \|fg\|_1 \) is close to \( \|f\|_p \|g\|_q \) precisely when the angle between \( \|f\|^{p/2} \) and \( \|g\|^{q/2} \) in \( L^2 \) is small, and the exponent 2 in that deviation term cannot be improved.<sup>[8](https://arxiv.org/abs/0710.2307)</sup>

Gangsong Leng and Yifan Lu later proved an optimal \( L^1 \) stability theorem in a discrete probabilistic form: for \( p > 1 \), \( q > 1 \) conjugate, and probability vectors \( a_k, b_k \ge 0 \) with \( \sum a_k = \sum b_k = 1 \),

\[ 1 - \sum_{k=1}^{n} a_k^{1/p} b_k^{1/q} \;\ge\; \frac{1}{2pq} \Bigl( \sum_{k=1}^{n} |a_k - b_k| \Bigr)^{2}, \]

and the constant \( 1/(2pq) \) is best possible; they also give the corresponding integral form. Their proof rests on a strengthened form of Young's inequality.<sup>[9](https://arxiv.org/abs/2605.31179)</sup> Stability estimates for Hölder's inequality also enter the Carlen–Frank–Lieb program on stability for the lowest eigenvalue of a Schrödinger operator, where the problem decomposes into a Hölder step and a Gagliardo–Nirenberg–Sobolev step.<sup>[10](https://arxiv.org/abs/2606.03749)</sup>

## History

The result traces to [Otto Hölder](https://www.edgepedia.org/otto-holder)'s 1889 paper "Ueber einen Mittelwerthssatz" (On a mean value theorem), published in the *Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen*, pages 38 to 47, and printed by the Dieterichsche Verlags-Buchhandlung in [Göttingen](https://www.edgepedia.org/gottingen).<sup>[3](https://de.wikisource.org/wiki/Ueber_einen_Mittelwerthssatz)</sup> The inequality was subsequently systematized in the classical treatise *Inequalities* by G. H. Hardy, J. E. Littlewood, and G. Pólya ([Cambridge University Press](https://www.edgepedia.org/cambridge-university-press), 1934).<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup> The Encyclopedia of Mathematics notes that the special case \( m = p = q = 2 \) was known earlier as the Bunyakovskii inequality and is called the Cauchy–Schwarz inequality in English-language literature.<sup>[1](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)</sup>

## References

1. [Hölder inequality, Encyclopedia of Mathematics (L. P. Kuptsov, originator)](https://encyclopediaofmath.org/wiki/H%C3%B6lder_inequality)
2. [18.100C/18.675 lecture notes, Real Analysis (Proposition 3.7.4, Hölder's Inequality), MIT](https://math.mit.edu/~setia/6710fa19.pdf)
3. [O. Hölder, "Ueber einen Mittelwerthssatz" (1889), Wikisource transcription](https://de.wikisource.org/wiki/Ueber_einen_Mittelwerthssatz)
4. [Analysis: Convergence and Duality, lecture notes on \( L^p \) spaces, Université de Bordeaux](https://www.math.u-bordeaux.fr/~pjaming/enseignement/CoursLp.pdf)
5. [Lecture notes for Math 222A, Sobolev spaces, University of California, Berkeley (Sung-Jin Oh)](https://web.math.berkeley.edu/~sjoh/pdfs/notes-math222a.pdf)
6. [Lecture 1: Sobolev spaces and their applications, MATH5021, The Chinese University of Hong Kong](https://www.math.cuhk.edu.hk/course_builder/2526/math5021/1-1%20Sobolev%20spaces.pdf)
7. [Christian D. Jäkel and Florian Robl, "The Hölder Inequality for KMS States", arXiv:1103.3608](https://arxiv.org/abs/1103.3608)
8. [J. M. Aldaz, "A stability version of Hölder's inequality", arXiv:0710.2307](https://arxiv.org/abs/0710.2307)
9. [Gangsong Leng and Yifan Lu, "An Optimal Stability Theorem for Hölder's Inequality", arXiv:2605.31179](https://arxiv.org/abs/2605.31179)
10. [Gangsong Leng, "A Density-Distance Version of the Carlen–Frank–Lieb Stability Theorem", arXiv:2606.03749](https://arxiv.org/abs/2606.03749)
11. [Measure Theory.part3.2015 02 05.pdf, uni-ulm.de](https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/sauter/ws14/measure-theory.part3.2015-02-05.pdf)

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