In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of [[Lp space| Lp spaces]]. The numbers p and q above are said to be Hölder conjugates of each other. The special case gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if fg1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in Lp(μ) and g is in Lq(μ) , then the pointwise product fg is in L1(μ) . Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space Lp(μ) , and also to establish that Lq(μ) is the dual space of Lp(μ) for p ∈ . 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.

Remarks

Conventions

The brief statement of Hölder's inequality uses some conventions. 1/∞ means zero. p, q ∈ , then fp and gq stand for the (possibly infinite) expressions , then f∞ stands for the essential supremum of , similarly for g∞ . fp with 1 ≤ p ≤ ∞ is a slight abuse, because in general it is only a norm of f if fp is finite and f is considered as equivalence class of μ-almost everywhere equal functions. If f ∈ Lp(μ) and g ∈ Lq(μ) , then the notation is adequate. 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 fg1 is finite, then the pointwise products of f with g and its complex conjugate function are μ-integrable, the estimate 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 L2(μ) , then Hölder's inequality for implies where the angle brackets refer to the inner product of L2(μ) . This is also called Cauchy–Schwarz inequality, but requires for its statement that f2 and g2 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 ) by using the functions and in place of f and g.

Generalization for probability measures

If (S, Σ, μ) is a probability space, then 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 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 with 1/p + 1/q = 1 .

Counting measure

For the n-dimensional Euclidean space, when the set S is with the counting measure, we have Often the following practical form of this is used, for any : For more than two sums, the following generalisation holds, with real positive exponents \lambda_i and : Equality holds iff. 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 complex-valued functions on S, then Hölder's inequality is

Probability measure

For the probability space let \mathbb{E} denote the expectation operator. For real- or complex-valued random variables X and Y on \Omega, Hölder's inequality reads Let and define Then is the Hölder conjugate of p. Applying Hölder's inequality to the random variables |X|^r and 1_{\Omega} we obtain In particular, if the sth absolute moment is finite, then the rth absolute moment is finite, too. (This also follows from Jensen's inequality.)

Product measure

For 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 denotes the product measure of μ1 and μ2 . 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 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 and 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 , 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 numbers α, 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 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. Alternative proof using Jensen's inequality: 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.

Extremal equality

Statement

Assume that 1 ≤ p < ∞ and let q denote the Hölder conjugate. Then for every f ∈ Lp(μ) , where max indicates that there actually is a g maximizing the right-hand side. When and if each set A in the σ-field Σ with contains a subset B ∈ Σ with 0 < μ(B) < ∞ (which is true in particular when μ is σ-finite), then Proof of the extremal equality:

Remarks and examples

Applications

f1 + f2p ≤ f1p + f2p for all f1 and f2 in Lp(μ) , see Minkowski inequality. f ∈ Lp(μ) defines a bounded (or continuous) linear functional κf on Lq(μ) by the formula κf as element of the continuous dual space Lq(μ)* coincides with the norm of f in Lp(μ) (see also the [[L^p-space| Lp -space]] article).

Generalization with more than two functions

Statement

Assume that r ∈ and p1, ..., pn ∈ such that where 1/∞ is interpreted as 0 in this equation. Then for all measurable real or complex-valued functions f1, ..., fn defined on S, 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 for all then Note: For 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 p1, ..., pn ∈ and let θ1, ..., θn ∈ (0, 1) denote weights with θ1 + ...

• θn = 1 . Define p as the weighted harmonic mean, that is, Given measurable real- or complex-valued functions f_k on S, then the above generalization of Hölder's inequality gives In particular, taking gives Specifying further and , in the case n = 2, we obtain the interpolation result An application of Hölder gives Both Littlewood and Lyapunov imply that if then f \in L^p for all

Reverse Hölder inequalities

Two functions

Assume that p ∈ (1, ∞) 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) ≠ 0 for μ-almost all s ∈ S , If then the reverse Hölder inequality is an equality if and only if Note: The expressions: and are not norms, they are just compact notations for

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, 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 be vectors with positive entries and such that for all i. If p,q,r are nonzero real numbers such that, then: 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 be n vectors in with positive entries and such that for all i. If are nonzero real numbers such that, then: As in the standard Hölder inequalities, there are corresponding statements for infinite sums and integrals.

Conditional Hölder inequality

Let (Ω,, \mathbb{P}) be a probability space, ⊂ a sub-σ-algebra, and p, q ∈ Hölder conjugates, meaning that 1/p + 1/q = 1 . Then for all real- or complex-valued random variables X and Y on Ω , Remarks: a > 0 with ∞ gives ∞. Proof of the conditional Hölder inequality:

Hölder's inequality for increasing seminorms

Let S be a set and let be the space of all complex-valued functions on S. Let N be an increasing seminorm on meaning that, for all real-valued functions we have the following implication (the seminorm is also allowed to attain the value ∞): Then: where the numbers p and q are Hölder conjugates. Remark: If (S, Σ, μ) is a measure space and N(f) is the upper Lebesgue integral of |f| then the restriction of N to all Σ -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.

Citations