Hölder's inequality 1

Hölder's inequalityIn mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental inequality betweenintegrals and an indispensable tool for the study of Lp 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- orcomplex-valued 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 formof the Cauchy–Schwarz inequality.Hölder's inequality holds even if ||fg ||1 is infinite, the right-hand side also being infinite in that case. In particular, if fis in Lp(μ) and g is in Lq(μ), then fg is in L1(μ).For 1 < p, q < ∞ and f ∈ Lp(μ) and g ∈ Lq(μ), Hölder's inequality becomes an equality if and only if |f |p and |g |q arelinearly dependent in L1(μ), meaning that there exist real numbers α, β ≥ 0, not both of them zero, such that α |f |p = β|g |q μ-almost everywhere.Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ), andalso to establish that Lq(μ) is the dual space of Lp(μ) for 1 ≤ p < ∞.Hölder's inequality was first found by L. J. Rogers (1888), and discovered independently by Hölder (1889).

Remarks

ConventionsThe 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 μ-almost everywhere equal functions. If f ∈ Lp(μ) and g ∈ Lq(μ), then thenotation is adequate.

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

Estimates for integrable productsAs above, let f and g denote measurable real- or complex-valued functions defined on S. If ||fg ||1 is finite, then theproducts of f with g and its complex conjugate function, respectively, are μ-integrable, the estimates

and the similar one for fg hold, and Hölder's inequality can be applied to the right-hand side. In particular, if f and gare in the Hilbert space L2(μ), then Hölder's inequality for p = q = 2 implies

Hölder's inequality 2

where the angle brackets refer to the inner product of L2(μ). This is also called Cauchy–Schwarz inequality, butrequires 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 measuresIf (S, Σ, μ) is a probability space, then 1 ≤ p, q ≤ ∞ just need to satisfy 1/p + 1/q ≤ 1, rather than being Hölderconjugates. A combination of Hölder's inequality and Jensen's inequality implies that

for all measurable real- or complex-valued functions f and g on S,

Notable special casesFor the following cases assume that p and q are in the open interval (1, ∞) with 1/p + 1/q = 1.

Counting measureIn the case of n-dimensional 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 measureIf S is a measurable subset of Rn with the Lebesgue measure, and f and g are measurable real- or complex-valuedfunctions on S, then Hölder inequality is

Probability measure

For the probability space , let E denote the expectation operator. For real- or complex-valued randomvariables 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 therandom variables |X |r and 1Ω, we obtain

In particular, if the sth absolute moment is finite, then the r th absolute moment is finite, too. (This also follows fromJensen's inequality.)

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Product measureFor two σ-finite measure spaces (S1, Σ1, μ1) and (S2, Σ2, μ2) define the product measure space by

where S is the Cartesian product of S1 and S2, the σ-algebra Σ arises as product σ-algebra of Σ1 and Σ2, and μ denotesthe product measure of μ1 and μ2. Then Tonelli's theorem allows us to rewrite Hölder's inequality using iteratedintegrals: If f and g are Σ-measurable real- or complex-valued functions on the Cartesian product S, then

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

Vector-valued functionsLet (S, Σ, μ) denote a σ-finite measure space and suppose that f = (f1, …, fn) and g = (g1, …, gn) are Σ-measurablefunctions on S, taking values in the n-dimensional real- or complex Euclidean space. By taking the product with thecounting measure on {1, …, n}, we can rewrite the above product measure version of Hölder's inequality in the form

If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbersα, β ≥ 0, not both of them zero, such that

    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 inequalityThere 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-handside 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 inthe following.If ||f ||p = ∞ or ||g ||q = ∞, then the right-hand 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 Hölder's inequality follows from the monotonicity ofthe 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

Hölder's inequality 4

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 existreal numbers α, β > 0, namely

   and    such that

   μ-almost everywhere   (*).The case ||f ||p = 0 corresponds to β = 0 in (*). The case ||g ||q = 0 corresponds to α = 0 in (*).

Extremal equality

StatementAssume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then, for every ƒ ∈ Lp(μ),

where max indicates that there actually is a g maximizing the right-hand side. When p = ∞ and if each set A in theσ-field Σ with μ(A) = ∞ contains a subset B ∈ Σ with 0 < μ(B) < ∞ (which is true in particular when μ is σ-finite),then

Remarks and examples• The equality for p = ∞ fails whenever there exists a set A in the σ-field Σ with μ(A) = ∞ that has no subset B ∈ Σ

with 0 < μ(B) < ∞ (the simplest example is the σ-field Σ containing just the empty set and S, and the measure μwith μ(S) = ∞). Then the indicator function 1A satisfies ||1A||∞ = 1, but every g ∈ L1(μ) has to be μ-almosteverywhere 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 1A 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 (withoutzero), 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 ∈ L1(μ) with 0 < ||g ||1 ≤ 1, let m denote the smallest natural number with g(m) ≠ 0. Then

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Applications• The extremal equality is one of the ways for proving the triangle inequality ||ƒ1 + ƒ2||p ≤ ||ƒ1||p + ||ƒ2||p for all ƒ1 and

ƒ2 in Lp(μ), see Minkowski inequality.• Hölder's inequality implies that every ƒ ∈ Lp(μ) defines a bounded (or continuous) linear functional κƒ on Lq(μ) by

the formula

The extremal equality (when true) shows that the norm of this functional κƒ as element of the continuous dualspace Lq(μ)∗ coincides with the norm of ƒ in Lp(μ) (see also the Lp-space article).

Generalization of Hölder's inequalityAssume that r ∈ (0, ∞) and p1, …, pn ∈ (0, ∞] such that

Then, for all measurable real- or complex-valued functions f1, …, fn 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 triangleinequality.

InterpolationLet p1, …, pn ∈ (0, ∞] and let θ1, …, θn ∈ (0, 1) denote weights with θ1+ … + θn = 1. Define p as the weightedharmonic mean, i.e.,

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

Then by the above generalization of Hölder's inequality,

In particular, taking θ1 = θ and θ2 = 1 - θ, in the case n =2, we obtain the interpolation result

for θ ∈ (0, 1) and

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Reverse Hölder inequalityAssume that p ∈ (1, ∞) and that the measure space (S, Σ, μ) satisfies μ(S) > 0. Then, for all measurable real- orcomplex-valued functions f and g on S such that g(s) ≠ 0 for μ-almost 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 ≥ 0such that

    μ-almost everywhere.Note: ||f ||1/p and ||g ||−1/(p −1) are not norms, these expressions are just compact notation for

   and   

Conditional Hölder inequalityLet be a probability space, a sub-σ-algebra, and p, q ∈ (1, ∞) Hölder conjugates, meaning that1/p + 1/q = 1. Then, for all real- or complex-valued random variables X and Y on Ω,

Remarks:

• If a non-negative random variable Z has infinite expected value, then its conditional expectation is defined by

• On the right-hand side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplyinga > 0 with ∞ gives ∞.

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" [1] (in German), Nachrichten von der Königl. Gesellschaft der

Wissenschaften und der Georg-Augusts-Universität zu Göttingen, Band 1889 (2): 38–47, JFM 21.0260.07.Available at Digi Zeitschriften [2].

• Kuptsov, L. P. (2001), "Hölder inequality" [3], 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" [4], Messenger of Mathematics,New Series XVII (10): 145–150, JFM 20.0254.02, archived from the original [5] on August 21, 2007.

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External links• Kuttler, Kenneth (2007), An introduction to linear algebra [6], Online e-book in PDF format, Brigham Young

University.• Lohwater, Arthur (1982) (PDF), Introduction to Inequalities [7].

References[1] http:/ / resolver. sub. uni-goettingen. de/ purl?GDZPPN00252421X[2] http:/ / www. digizeitschriften. de/ index. php?id=64& L=2[3] http:/ / www. encyclopediaofmath. org/ index. php?title=H/ h047514[4] http:/ / www. archive. org/ stream/ messengermathem01unkngoog#page/ n183/ mode/ 1up[5] http:/ / www. archive. org/ details/ messengermathem01unkngoog[6] http:/ / www. math. byu. edu/ ~klkuttle/ Linearalgebra. pdf[7] http:/ / www. mediafire. com/ ?1mw1tkgozzu

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