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.
Let (S, Σ, μ) be a measure space and let 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1. Then, for all measurable real or complexvalued functions f and g on S,
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 righthand side also being infinite in that case. In particular, if f is in L^{p}(μ) and g is in L^{q}(μ), then fg is in L^{1}(μ).
For 1 < p, q < ∞ and f ∈ L^{p}(μ) and g ∈ L^{q}(μ), 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 ≥ 0, not both of them zero, such that α f ^{p} = β g ^{q} μalmost everywhere.
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 1 ≤ p < ∞.
Hölder's inequality was first found by L. J. Rogers (1888), and discovered independently by Hölder (1889).
Contents
Remarks
Conventions
The brief statement of Hölder's inequality uses some conventions.
 In the definition of Hölder conjugates, 1/ ∞ means zero.
 If 1 ≤ p, q < ∞, then f _{p} and g _{q} stand for the (possibly infinite) expressions

 and
 If p = ∞, then f _{∞} stands for the essential supremum of f , similarly for g _{∞}.
 The notation f _{p} with 1 ≤ p ≤ ∞ 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 μalmost everywhere equal functions. If f ∈ L^{p}(μ) and g ∈ L^{q}(μ), then the notation is adequate.
 On the righthand side of Hölder's inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying a > 0 with ∞ gives ∞.
Estimates for integrable products
As above, let f and g denote measurable real or complexvalued functions defined on S. If fg _{1} is finite, then the products of f with g and its complex conjugate function, respectively, are μintegrable, the estimates
and the similar one for fg hold, and Hölder's inequality can be applied to the righthand side. In particular, if f and g are in the Hilbert space L^{2}(μ), then Hölder's inequality for p = q = 2 implies
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.
Generalization for probability measures
If (S, Σ, μ) is a probability space, then 1 ≤ p, q ≤ ∞ just need to satisfy 1/p + 1/q ≤ 1, rather than being Hölder conjugates. A combination of Hölder's inequality and Jensen's inequality implies that
for all measurable real or complexvalued functions f and g on S,
Notable special cases
For the following cases assume that p and q are in the open interval (1, ∞) with 1/p + 1/q = 1.
Counting measure
In the case of ndimensional Euclidean space, when the set S is {1, …, n} with the counting measure, we have
If S = N with the counting measure, then we get Hölder's inequality for sequence spaces:
Lebesgue measure
If S is a measurable subset of R^{n} with the Lebesgue measure, and f and g are measurable real or complexvalued functions on S, then Hölder inequality is
Probability measure
For the probability space , let E denote the expectation operator. For real or complexvalued random variables X and Y on Ω, Hölder's inequality reads
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
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 σfinite measure spaces (S_{1}, Σ_{1}, μ_{1}) and (S_{2}, Σ_{2}, μ_{2}) define the product measure space by
where S is the Cartesian product of S_{1} and S_{2}, the σalgebra Σ arises as product σalgebra 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 Σmeasurable real or complexvalued functions on the Cartesian product S, then
This can be generalized to more than two σfinite measure spaces.
Vectorvalued functions
Let (S, Σ, μ) denote a σfinite measure space and suppose that f = (f_{1}, …, f_{n}) and g = (g_{1}, …, g_{n}) are Σmeasurable functions on S, taking values in the ndimensional 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
This 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 μalmost everywhere, and the product fg is zero μalmost everywhere, hence the lefthand 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} = ∞ or g _{q} = ∞, then the righthand side of Hölder's inequality is infinite. Therefore, we may assume that f _{p} and g _{q} are in (0, ∞).
If p = ∞ and q = 1, then fg  ≤ f _{∞} g  almost everywhere and Hölders inequality follows from the monotonicity of the Lebesgue integral. Similarly for p = 1 and q = ∞. Therefore, we may also assume p, q ∈ (1, ∞).
Dividing f and g by f _{p} and g _{q}, respectively, we can assume that
We now use Young's inequality, which states that
for all nonnegative a and b, where equality is achieved if and only if a ^{p} = b ^{q}. Hence
Integrating both sides gives
which proves the claim.
Under the assumptions p ∈ (1, ∞) and f _{p} = g _{q} = 1, equality holds if and only if f ^{p} = g ^{q} almost everywhere. More generally, if f _{p} and g _{q} are in (0, ∞), then Hölder's inequality becomes an equality if and only if there exist real numbers α, β > 0, namely
 and
such that
 μalmost everywhere (*)
The case f _{p} = 0 corresponds to β = 0 in (*). The case g _{q} = 0 corresponds to α = 0 in (*).
Extremal equality
Statement
Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then, for every ƒ ∈ L^{p}(μ),
where max indicates that there actually is a g maximizing the righthand side. When p = ∞ and if each set A in the σfield Σ with μ(A) = ∞ contains a subset B ∈ Σ with 0 < μ(B) < ∞ (which is true in particular when μ is σfinite), then
Proof of the extremal equalityRemarks and examples
 The equality for p = ∞ fails whenever there exists a set A in the σfield Σ with μ(A) = ∞ that has no subset B ∈ Σ with 0 < μ(B) < ∞ (the simplest example is the σfield Σ containing just the empty set and S, and the measure μ with μ(S) = ∞). Then the indicator function 1_{A} satisfies 1_{A}_{∞} = 1, but every g ∈ L^{1}(μ) has to be μalmost everywhere constant on A, because it is Σmeasurable, and this constant has to be zero, because g is μintegrable. Therefore, the above supremum for the indicator function 1_{A} is zero and the extremal equality fails.
 For p = ∞, 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 ƒ(n) = (n − 1)/n for every natural number n. Then ƒ _{∞} = 1. For g ∈ L^{1}(μ) with 0 < g _{1} ≤ 1, let m denote the smallest natural number with g(m) ≠ 0. Then
Applications
 The extremal equality is one of the ways for proving the triangle inequality ƒ_{1} + ƒ_{2}_{p} ≤ ƒ_{1}_{p} + ƒ_{2}_{p} for all ƒ_{1} and ƒ_{2} in L^{p}(μ), see Minkowski inequality.
 Hölder's inequality implies that every ƒ ∈ L^{p}(μ) defines a bounded (or continuous) linear functional κ_{ƒ} on L^{q}(μ) by the formula

 The extremal equality (when true) shows that the norm of this functional κ_{ƒ} as element of the continuous dual space L^{q}(μ)^{∗} coincides with the norm of ƒ in L^{p}(μ) (see also the L^{p}space article).
Generalization of Hölder's inequality
Assume that r ∈ (0, ∞) and p_{1}, …, p_{n} ∈ (0, ∞] such that
Then, for all measurable real or complexvalued functions f_{1}, …, f_{n} defined on S,
In particular,
Note:
 For r ∈ (0, 1), contrary to the notation, ._{r} is in general not a norm, because it doesn't satisfy the triangle inequality.
Proof of the generalizationInterpolation
Let p_{1}, …, p_{n} ∈ (0, ∞] and let θ_{1}, …, θ_{n} ∈ (0, 1) denote weights with θ_{1}+ … + θ_{n} = 1. Define p as the weighted harmonic mean, i.e.,
Given a measurable real or complexvalued functions f on S, define
Then by the above generalization of Hölder's inequality,
As an interpolation result^{[citation needed]} for n = 2,
where θ ∈ (0, 1) and
Reverse Hölder inequality
Assume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then, for all measurable real or complexvalued functions f and g on S such that g(s) ≠ 0 for μalmost all s ∈ S,
If fg _{1} < ∞ and g _{−1/(p −1)} > 0, then the reverse Hölder inequality is an equality if and only if there exists an α ≥ 0 such that
 μalmost everywhere.
Note: f _{1/p} and g _{−1/(p −1)} are not norms, these expressions are just compact notation for
 and
Proof of the reverse Hölder inequalityConditional Hölder inequality
Let be a probability space, a subσalgebra, and p, q ∈ (1, ∞) Hölder conjugates, meaning that 1/p + 1/q = 1. Then, for all real or complexvalued random variables X and Y on Ω,
Remarks:
 If a nonnegative random variable Z has infinite expected value, then its conditional expectation is defined by
 On the righthand 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 inequalityReferences
 Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1934), Inequalities, Cambridge University Press, pp. XII+314, ISBN 0521358809, JFM 60.0169.01, Zbl 0010.10703.
 Hölder, O. (1889), "Ueber einen Mittelwertsatz" (in German), Nachrichten von der Königl. Gesellschaft der Wissenschaften und der GeorgAugustsUniversität zu Göttingen, Band 1889 (2): 38–47, JFM 21.0260.07, http://resolver.sub.unigoettingen.de/purl?GDZPPN00252421X. Available at Digi Zeitschriften.
 Kuptsov, L. P. (2001), "Hölder inequality", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/H/h047514.htm.
 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, http://www.archive.org/stream/messengermathem01unkngoog#page/n183/mode/1up.
External links
 Kuttler, Kenneth (2007), An introduction to linear algebra, Online ebook in PDF format, Brigham Young University, http://www.math.byu.edu/~klkuttle/Linearalgebra.pdf.
 Lohwater, Arthur (1982), Introduction to Inequalities, Online ebook in PDF fomat, http://www.mediafire.com/?1mw1tkgozzu.
Categories: Inequalities
 Functional analysis
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