Hölder's inequality explained

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

\begin{align} &\left(\int _{S} |f| ^{pd\mu\right)} ^{(1)/(p)} \\ &\left(\int _{S} |g| ^{qd\mu\right)} ^{(1)/(q)} \end{align}

• 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

l |\int _{S} f\bar g d \mu r |\le\int _{S|fg|d\mu} = \|fg\| _{1}

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

|\langle f,g\rangle| \le \|f\| _{2} \|g\| _{2,}

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

\|fg\| _{1} \le \|f\| _{p} \|g\| _{q}

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

n -dimensional Euclidean space, when the set S is \{ 1 , ... ,n\} with the counting measure, we have\sum ^{n} _{k =1} |x _{ky} _{k|} \le l ( \sum ^{n} _{k =1} |x ^{p} _{k|} r ) ^{(1)/(p)} l ( \sum ^{n} _{k =1} |y ^{q} _{k|} r ) ^{(1)/(q)} forall (x _{1,\ldots,x} _{n),(y} _{1,\ldots,y} ^{n or C} ^{n)\in R} ^{n.} Often the following practical form of this is used, for any (r,s)\in R _{+} :l (\sum ^{n} _{k =1} |x ^{s} _{k|} r ) ^{r+s} \le l ( \sum ^{n} _{k =1} |x ^{r + s} _{k|} r ) ^{r} l ( \sum ^{n} _{k =1} |y ^{r + s} _{k|} r ) ^{s} . For more than two sums, the following generalisation holds, with real positive exponents λ _{i} and λ _{a} + λ _{b+} … +λ _{z} =1 :\sum ^{n} _{k =1} |a ^{λ _{a}} _{k|} |b ^{λ _{b}} _{k|} … |z ^{λ _{z}} _{k|} \le l (\sum ^{n} _{k =1} |a ^{λ _{a}} _{k| r )} l (\sum ^{n} _{k =1} |b ^{λ _{b}} _{k| r )} … l (\sum ^{n} _{k =1} |z ^{λ _{z}} _{k| r )} . Equality holds iff |a _{1|:} |a _{2|:} … : |a _{n|} = |b _{1|:} |b _{2|:} … : |b _{n|} = … = |z _{1|:} |z _{2|:} … : |z _{n|} .

If

S = \N with the counting measure, then we get Hölder's inequality for sequence spaces:\sum ^{inf ty} _{k =1} |x _{ky} _{k|} \le l ( \sum ^{inf ty} _{k =1} |x ^{p} _{k|} r ) ^{(1)/(p)} \left( \sum ^{inf ty} _{k =1} |y ^{q} _{k|} \right) ^{(1)/(q)} forall (x _{k)} _{k\inN} , (y _{k)} _{k\inN} \in R ^{N} or C ^{N} .

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\int _{S} l | f(x)g(x) r | d x \le l (\int _{S} |f(x)| ^{pdxr)} ^{(1)/(p)} l (\int _{S} |g(x)| ^{qdxr)} ^{(1)/(q)} .

Probability measure

(\Omega, l{F}, P ), let E denote the expectation operator. For real- or complex-valued random variables X and Y on \Omega, Hölder's inequality readsE [|XY|] \leqslant \left ( E l [ |X| ^{pr]\right)} ^{(1)/(p)} \left( E l [|Y| ^{qr]\right)} ^{(1)/(q)} .

Let

1 < r < s < inf ty and define p = \tfrac{s}{r}. Then q = \tfrac{p}{p -1 } is the Hölder conjugate of p. Applying Hölder's inequality to the random variables |X| ^{r} and 1 _{\Omega} we obtainE l [|X| ^{rr]\leqslant} \left( E l [|X| ^{sr]\right)} ^{(r)/(s)} .

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

S = S _{1 x} S _{2,} \Sigma = \Sigma _{1 ⊗ \Sigma} _{2,} \mu = \mu _{1 ⊗ \mu} _{2,}

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

\int _{S _{1}} \int _{S _{2}} |f(x,y)g(x,y)|\mu _{2(dy)\mu} _{1(dx)} \le\left(\int _{S _{1}} \int _{S _{2}} |f(x,y)| ^{p\mu} _{2 ( d y)\mu} ^{(1)/(p)} ^{1 ( d x)\right)} \left(\int _{S _{1}} \int _{S _{2}} |g(x,y)| ^{q\mu} _{2 ( d y)\mu} ^{(1)/(q)} ^{1 ( d x)\right)} .

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

\int _{S} \sum ^{n|f} _{k(x)g} _{k(x)|\mu(dx)} \le \left(\int _{S\sum} ^{p\mu( d x)\right)} ^{k(x)|} ^{(1)/(p)} \left(\int _{S\sum} ^{q\mu( d x)\right)} ^{k(x)|} ^{(1)/(q)} .

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

\alpha \left (|f ^{p} _{n(x)|} \right ) = \beta \left (|g ^{q} _{n(x)|} \right ),

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,

\|f\| _{p} = max \left \{ \left| \int _{S} f g d \mu \right | : g\in L ^{q(\mu),} \|g\| _{q} \le 1 \right\},

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

\|f\| _{infty} = \sup \left\{ \left| \int _{S} f g d \mu \right| : g\in L ^{1(\mu),} \|g\| _{1} \le 1 \right \}. Proof of the extremal equality:

Remarks and examples

• The equality for

p = inf ty fails whenever there exists a set A of infinite measure in the \sigma -field \Sigma with that has no subset B \in \Sigma that satisfies: 0 < \mu(B) < inf ty. (the simplest example is the \sigma -field \Sigma containing just the empty set and S, and the measure \mu with \mu(S) = inf ty. ) Then the indicator function 1 _{A} satisfies \| 1 _{A\|} _{infty} = 1 , but every g \in L ^{1} (\mu) has to be \mu -almost everywhere constant on A, because it is \Sigma -measurable, and this constant has to be zero, because g is \mu -integrable. Therefore, the above supremum for the indicator function 1 _{A} is zero and the extremal equality fails.

• For

p = inf ty, the supremum is in general not attained. As an example, let S = N , \Sigma = l{P}( N ) and \mu the counting measure. Define:\begin{cases} f: N \to R \ f(n) = (n -1)/(n) \end{cases}

Then

\|f\| _{infty} = 1 . For g \in L ^{1} (\mu, N ) with 0 < \|g\| _{1} \leqslant 1 , let m denote the smallest natural number with g(m) ≠ 0 . Then\left |\int _{S} fg d \mu\right| \leqslant (m -1)/(m) |g(m)| + \sum ^{inf ty|g(n)|} _{n = m +1} = \|g\| _{1- (|g(m)|)/(m< 1 .)}

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

\kappa _{f(g)} = \int _{S} f g d \mu, g\in L ^{q(\mu).}

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

\sum ^{n} _{k =1} (1 {p)/(_{k}}) = (1 r)/()

where 1/∞ is interpreted as 0 in this equation. Then for all measurable real or complex-valued functions defined on,

\left\|\prod ^{n} _{k =1} f _{k\right\|} _{r} \le \prod ^{n} _{k =1} \left\|f _{k\right\|} ^{p _{k}}

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

f _{k} \in L ^{p _{k}} (\mu) for all k \in \{ 1 , \ldots, n \} then \prod ^{n} _{k =1} f _{k} \in L ^{r(\mu).}

Note: For

r \in ( 0 , 1 ), 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

p as the weighted harmonic mean, that is,(1 p)/(=) \sum ^{n} _{k =1} (\theta _{k})/(p _{k}) .

Given measurable real- or complex-valued functions

f _{k} on, then the above generalization of Hölder's inequality gives\left\| |f ^{\theta _{1}} _{1 |} … |f ^{\theta _{n}} _{n|} \right\| _{p} \le \left\||f ^{\theta _{1}} _{1 |} \right\| _{(p _{1})/(\theta _{1})} … \left\| |f ^{\theta _{n}} _{n|} \right\| _{(p _{n})/(\theta _{n})} = \|f _{1\|} ^{\theta _{1}} ^{p _{1}} … \|f _{n\|} ^{\theta _{n}} ^{p _{n}} .

In particular, taking

f _{1} = … = f _{n=:f} gives\|f\| _{p} \leqslant \prod ^{n} _{k =1} \|f\| ^{\theta _{k}} _{p _{k}} .

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

f \in L ^{p _{0}} \cap L ^{p _{1}} then f \in L ^{p} for all p _{0} < p < p _{1.} ^[1]

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,

\|fg\| _{1\geqslant} \|f\| _{(1)/(p)} \|g\| _{(-1)/(p -1)} .

If

\|fg\| _{1} < inf ty and \|g\| _{(-1)/(p -1)} > 0 ,

then the reverse Hölder inequality is an equality if and only if

\exists \alpha \geqslant 0 |f| = \alpha|g| ^{(- p)/(p -1)} \mu -almosteverywhere .

Note: The expressions:

\|f\| _{(1)/(p)} and \|g\| _{(-1)/(p -1)} ,

are not norms, they are just compact notations for

\left (\int ^{(1)/(p)} _{S|f|} d \mu\right) ^{p} and \left (\int ^{(-1)/(p -1)} _{S|g|} d \mu\right) ^{-(p-1)} .

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

p _{1,...,} p _{m-1} < 0 and p _{m} \in R be such that \sum ^{m} _{k =1} (1)/(p _{k}) = 1 (hence 0 < p _{m} < 1 ). Let f _{k} be measurable functions for k = 1 ,...,m . Then\left\|\prod ^{m} _{k =1} f _{k\right\|} _{1} \ge \prod ^{m} _{k =1} \left\|f _{k\right\|} ^{p _{k}} .

This follows from the symmetric form of the Hölder inequality (see below).

Symmetric forms of Hölder inequality

It was observed by Aczél and Beckenbach^[2] that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function):

Let

f = (f( 1 ), ... ,f(m)) , g = (g( 1 ), ... , g(m)), h = (h( 1 ), ... ,h(m)) be vectors with positive entries and such that f(i) g(i) h(i) = 1 for all i . If p,q,r are nonzero real numbers such that (1)/(+ p) (1)/(+ q) (1)/(r) =0 , then:\|f\| _{p} \|g\| _{q} \|h\| _{r} \ge 1 if all but one of p,q,r are positive;\|f\| _{p} \|g\| _{q} \|h\| _{r} \le 1 if all but one of p,q,r 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

f _{1,} ... , f _{n} be n vectors in R ^{m} with positive entries and such that f _{1(i)} ... f _{n(i)} = 1 for all i . If p _{1,...,p} _{n} are nonzero real numbers such that (1)/(+...+ p _{1}) (1)/(p _{n}) =0 , then:\|f _{1\|} ^{p _{1}} ... \|f _{n\|} ^{p _{n}} \ge 1 if all but one of the numbers p _{i} are positive;\|f _{1\|} ^{p _{1}} ... \|f _{n\|} ^{p _{n}} \le 1 if all but one of the numbers p _{i} 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 ,

E l [|XY| | l{G} r ] \le l ( E l [|X| ^{p|l{G}r]r)} ^{(1)/(p)} l ( E l [|Y| ^{q|l{G}r]r)} ^{(1)/(q)} P -almostsurely.

Remarks:

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

E [Z| l{G}] = \sup _{n\inN} E [ min \{Z,n\}| l{G}] a.s.

• 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

F(S, C ) be the space of all complex-valued functions on . Let be an increasing seminorm on F(S, C ), meaning that, for all real-valued functions f, g \in F(S, C ) we have the following implication (the seminorm is also allowed to attain the value ∞):\forall s \in S f(s) \geqslant g(s) \geqslant 0 ⇒ N(f) \geqslant N(g).

Then:

\forall f, g \in F(S, C ) N(|fg|) \leqslant l (N(|f| ^{p)r)} ^{(1)/(p)} l (N(|g| ^{q)r)} ^{(1)/(q)} ,

where the numbers

p and q are Hölder conjugates.^[3]

Remark: If is a measure space and

N(f) is the upper Lebesgue integral of |f| 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^[4] 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

References

• • .
• . Available at Digi Zeitschriften.
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External links

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• Archived at Ghostarchive and the Wayback Machine: .

Notes and References

1. Book: Wojtaszczyk, P. . Banach Spaces for Analysts . 1991 . Cambridge University Press . 978-0-521-56675-9 . Cambridge Studies in Advanced Mathematics . Cambridge.
2. Book: Beckenbach . E. F. . General inequalities 2 . International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d'Analyse Numérique . 1980 . 47 . Birkhäuser Basel . 978-3-7643-1056-1 . 145–150 . 10.1007/978-3-0348-6324-7 .
3. For a proof see .
4. Nielsen . Frank . Sun . Ke . Marchand-Maillet . Stephane. On Hölder projective divergences. 2017 . 3. 19 . Entropy . 122 . 10.3390/e19030122 . 1701.03916 . 2017Entrp..19..122N . free .