In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces.
The numbers and 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 is infinite, the right-hand side also being infinite in that case. Conversely, if is in and is in , then the pointwise product is in .
Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space , and also to establish that is the dual space of for .
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.
In the definition of Hölder conjugates, means zero. If , then and stand for the (possibly infinite) expressions
If , then stands for the essential supremum of , similarly for . The notation with is a slight abuse, because in general it is only a norm of if is finite and is considered as equivalence class of -almost everywhere equal functions. If and , then the notation is adequate. On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means 0. Multiplying with ∞ gives ∞.
Estimates for integrable products As above, let and denote measurable real- or complex-valued functions defined on . If is finite, then the pointwise products of with and its complex conjugate function are -integrable, the estimate
and the similar one for hold, and Hölder's inequality can be applied to the right-hand side. In particular, if and are in the Hilbert space , then Hölder's inequality for implies
where the angle brackets refer to the inner product of . This is also called Cauchy–Schwarz inequality, but requires for its statement that and are finite to make sure that the inner product of and is well defined. We may recover the original inequality (for the case ) by using the functions and in place of and .
Generalization for probability measures If is a probability space, then just need to satisfy , 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 and on .
Notable special cases For the following cases assume that and are in the open interval with .
Counting measure For the -dimensional Euclidean space, when the set 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 and :
Equality holds iff .
If with the counting measure, then we get Hölder's inequality for sequence spaces:
Lebesgue measure If is a measurable subset of with the Lebesgue measure, and and are measurable real- or complex-valued functions on , then Hölder's inequality is
Probability measure For the probability space let denote the expectation operator. For real- or complex-valued random variables and on Hölder's inequality reads
Let and define Then is the Hölder conjugate of Applying Hölder's inequality to the random variables and we obtain
In particular, if the th absolute moment is finite, then the th absolute moment is finite, too. (This also follows from Jensen's inequality.)
Product measure For two σ-finite measure spaces and define the product measure space by
where is the Cartesian product of and , the arises as product σ-algebra of and , and denotes the product measure of and . Then Tonelli's theorem allows us to rewrite Hölder's inequality using iterated integrals: If and are real- or complex-valued functions on the Cartesian product , then
This can be generalized to more than two measure spaces.
Vector-valued functions Let denote a measure space and suppose that and are -measurable functions on , taking values in the -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 , not both of them zero, such that
for -almost all in .
This finite-dimensional version generalizes to functions and 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 and let denote the Hölder conjugate. Then for every ,
where max indicates that there actually is a maximizing the right-hand side. When and if each set in the with contains a subset with (which is true in particular when is ), then
Proof of the extremal equality:
Remarks and examples
The equality for fails whenever there exists a set of infinite measure in the -field with that has no subset that satisfies: (the simplest example is the -field containing just the empty set and and the measure with ) Then the indicator function satisfies but every has to be -almost everywhere constant on because it is -measurable, and this constant has to be zero, because is -integrable. Therefore, the above supremum for the indicator function is zero and the extremal equality fails. For the supremum is in general not attained. As an example, let and the counting measure. Define:
Then For with let denote the smallest natural number with Then
Applications The extremal equality is one of the ways for proving the triangle inequality for all and in , see Minkowski inequality. Hölder's inequality implies that every defines a bounded (or continuous) linear functional on by the formula
The extremal equality (when true) shows that the norm of this functional as element of the continuous dual space coincides with the norm of in (see also the article).
Generalization with more than two functions
Statement Assume that and such that
where 1/∞ is interpreted as 0 in this equation, and r=∞ implies are all equal to ∞. Then, for all measurable real or complex-valued functions defined on ,
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, is in general not a norm because it doesn't satisfy the triangle inequality.
Proof of the generalization:
Interpolation Let and let denote weights with . Define as the weighted harmonic mean, that is,
Given measurable real- or complex-valued functions on , then the above generalization of Hölder's inequality gives
In particular, taking gives
Specifying further and , in the case we obtain the interpolation result
An application of Hölder gives
Both Littlewood and Lyapunov imply that if then for all
Reverse Hölder inequalities
Two functions Assume that and that the measure space satisfies . Then for all measurable real- or complex-valued functions and on such that for all ,
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,
Let and be such that (hence ). Let be measurable functions for . Then
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 . If are nonzero real numbers such that , then: if all but one of are positive; if all but one of are negative.
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 vectors in with positive entries and such that for all . If are nonzero real numbers such that , then: if all but one of the numbers are positive; if all but one of the numbers are negative.
As in the standard Hölder inequalities, there are corresponding statements for infinite sums and integrals.
Conditional Hölder inequality Let be a probability space, a , and Hölder conjugates, meaning that . Then for all real- or complex-valued random variables and on ,
Remarks: If a non-negative random variable 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. Multiplying with ∞ gives ∞.
Proof of the conditional Hölder inequality:
Hölder's inequality for increasing seminorms Let be a set and let be the space of all complex-valued functions on . Let 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 and are Hölder conjugates.For a proof see .
Remark: If is a measure space and is the upper Lebesgue integral of then the restriction of to all 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.
See also Cauchy–Schwarz inequality Minkowski inequality Jensen's inequality Young's inequality for products Clarkson's inequalities Brascamp–Lieb inequality
Citations
References
. . Available at Digi Zeitschriften. .
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