View source for Hölder's inequality

{{Short description|Inequality between integrals in Lp spaces}}
In [[mathematical analysis]], '''Hölder's inequality''', named after [[Otto Hölder]], is a fundamental [[inequality (mathematics)|inequality]] between [[Lebesgue integration|integrals]] and an indispensable tool for the study of [[Lp space|{{math|''L<sup>p</sup>''}} spaces]].

{{Math theorem
| name = Hölder's inequality
| note = 
| math_statement = 
Let {{math|(''S'', Σ, ''μ'')}} be a [[measure space]] and let {{math|''p'', ''q'' ∈}} {{closed-closed|1, ∞}} with {{math|1/''p'' + 1/''q'' {{=}} 1}}. Then for all [[measurable function|measurable]] [[real number|real]]- or [[complex number|complex]]-valued [[function (mathematics)|functions]] {{mvar|f}} and {{mvar|g}} on {{mvar|S}},

<math display="block">\|fg\|_1 \le \|f\|_p \|g\|_q.</math>

If, in addition, {{math|''p'', ''q'' ∈}} {{open-open|1, ∞}} and {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}} and {{math|''g'' ∈ ''L<sup>q</sup>''(''μ'')}}, then Hölder's inequality becomes an equality if and only if {{math|{{abs|''f''{{hair space}}}}<sup>''p''</sup>}} and {{math|{{!}}''g''{{!}}<sup>''q''</sup>}} are [[Linear dependence|linearly dependent]] in {{math|''L''<sup>1</sup>(''μ'')}}, meaning that there exist real numbers {{math|''α'', ''β'' ≥ 0}}, not both of them zero, such that {{math|''α''{{!}}''f''{{hair space}}{{!}}<sup>''p''</sup> {{=}} ''β'' {{!}}''g''{{!}}<sup>''q''</sup>}} {{mvar|μ}}-[[almost everywhere]].
}}

The numbers {{mvar|p}} and {{mvar|q}} above are said to be '''[[Hölder conjugates]]''' of each other. The special case {{math|''p'' {{=}} ''q'' {{=}} 2}} gives a form of the [[Cauchy–Schwarz inequality]].<ref>{{harvnb|Roman|2008|loc=p. 303 §12}}</ref> Hölder's inequality holds even if {{math|{{norm|''fg''}}<sub>1</sub>}} is [[Infinity|infinite]], the right-hand side also being infinite in that case. Conversely, if {{mvar|f}} is in {{math|''L<sup>p</sup>''(''μ'')}} and {{mvar|g}} is in {{math|''L<sup>q</sup>''(''μ'')}}, then the [[pointwise product]] {{math|''fg''}} is in {{math|''L''<sup>1</sup>(''μ'')}}.

Hölder's inequality is used to prove the [[Minkowski inequality]], which is the [[triangle inequality]] in the space {{math|''L<sup>p</sup>''(''μ'')}}, and also to establish that {{math|''L<sup>q</sup>''(''μ'')}} is the [[dual space]] of {{math|''L<sup>p</sup>''(''μ'')}} for {{math|''p'' ∈}} {{closed-open|1, ∞}}.

Hölder's inequality (in a slightly different form) was first found by {{harvs|first=Leonard James|last=Rogers|author-link=Leonard James Rogers|txt|year=1888}}. Inspired by Rogers' work, {{harvtxt|Hölder|1889}} gave another proof as part of a work developing the concept of [[convex function|convex and concave functions]] and introducing [[Jensen's inequality]],<ref>{{citation
 | last = Maligranda | first = Lech
 | doi = 10.7153/mia-01-05
 | issue = 1
 | journal = Mathematical Inequalities & Applications
 | mr = 1492911
 | pages = 69–83
 | title = Why Hölder's inequality should be called Rogers' inequality
 | volume = 1
 | year = 1998| doi-access = free
 }}</ref> which was in turn named for work of [[Johan Jensen (mathematician)|Johan Jensen]] building on Hölder's work.<ref>{{citation
 | last1 = Guessab | first1 = A.
 | last2 = Schmeisser | first2 = G.
 | doi = 10.1007/s00013-013-0522-3
 | issue = 6
 | journal = Archiv der Mathematik
 | mr = 3069109
 | quote = under the additional assumption that <math>\varphi''</math> exists, this inequality was already obtained by Hölder in 1889
 | pages = 561–570
 | title = Necessary and sufficient conditions for the validity of Jensen's inequality
 | volume = 100
 | year = 2013| s2cid = 253600514
 }}</ref>

==Remarks==

===Conventions===
The brief statement of Hölder's inequality uses some conventions.

* In the definition of Hölder conjugates, {{math|1/∞}} means zero.
* If {{math|''p'', ''q'' ∈}} {{closed-open|1, ∞}}, then {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} and {{math|{{norm|''g''}}<sub>''q''</sub>}} stand for the (possibly infinite) expressions
::<math>\begin{align}
&\left(\int_S |f|^p\,\mathrm{d}\mu\right)^{\frac{1}{p}} \\
&\left(\int_S |g|^q\,\mathrm{d}\mu\right)^{\frac{1}{q}}
\end{align}</math>

* If {{math|''p'' {{=}} ∞}}, then {{math|{{norm|''f''{{hair space}}}}<sub>∞</sub>}} stands for the [[essential supremum]] of {{math|{{abs|''f''{{hair space}}}}}}, similarly for {{math|{{norm|''g''}}<sub>∞</sub>}}.
* The notation {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} with {{math|1 ≤ ''p'' ≤ ∞}} is a slight abuse, because in general it is only a [[norm (mathematics)|norm]] of {{mvar|f}} if {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} is finite and {{mvar|f}} is considered as [[equivalence class]] of {{mvar|μ}}-almost everywhere equal functions. If  {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}} and {{math|''g'' ∈ ''L<sup>q</sup>''(''μ'')}}, then the notation is adequate.
* On the right-hand side of Hölder's inequality, 0 × ∞ as well as ∞ × 0 means&nbsp;0. Multiplying {{math|''a'' > 0}} with ∞ gives&nbsp;∞.

===Estimates for integrable products===
As above, let {{mvar|f}} and {{mvar|g}} denote measurable real- or complex-valued functions defined on {{mvar|S}}. If {{math|{{norm|''fg''}}<sub>1</sub>}} is finite, then the pointwise products of {{mvar|f}} with {{mvar|g}} and its [[complex conjugate]] function are {{mvar|μ}}-integrable, the estimate

:<math>\biggl|\int_S f\bar g\,\mathrm{d}\mu\biggr|\le\int_S|fg|\,\mathrm{d}\mu =\|fg\|_1</math>

and the similar one for {{math|''fg''}} hold, and Hölder's inequality can be applied to the right-hand side. In particular, if {{mvar|f}} and {{mvar|g}} are in the [[Hilbert space]] {{math|''L''<sup>2</sup>(''μ'')}}, then Hölder's inequality for {{math|''p'' {{=}} ''q'' {{=}} 2}} implies

:<math>|\langle f,g\rangle| \le \|f\|_2 \|g\|_2,</math>

where the angle brackets refer to the [[inner product]] of {{math|''L''<sup>2</sup>(''μ'')}}. This is also called [[Cauchy–Schwarz inequality]], but requires for its statement that {{math|{{norm|''f''{{hair space}}}}<sub>2</sub>}} and {{math|{{norm|''g''}}<sub>2</sub>}} are finite to make sure that the inner product of {{mvar|f}} and {{mvar|g}} is well defined. We may recover the original inequality (for the case {{math|''p'' {{=}} 2}}) by using the functions {{math|{{abs|''f''{{hair space}}}}}} and {{math|{{abs|''g''}}}} in place of {{mvar|f}} and {{mvar|g}}.

===Generalization for probability measures===
If {{math|(''S'', Σ, ''μ'')}} is a [[probability space]], then {{math|''p'', ''q'' ∈}} {{closed-closed|1, ∞}} just need to satisfy {{math|1/''p'' + 1/''q'' ≤ 1}}, rather than being Hölder conjugates. A combination of Hölder's inequality and [[Jensen's inequality]] implies that

:<math>\|fg\|_1 \le \|f\|_p \|g\|_q</math>

for all measurable real- or complex-valued functions {{mvar|f}} and {{mvar|g}} on&nbsp;{{mvar|S}}.

==Notable special cases==
For the following cases assume that {{mvar|p}} and {{mvar|q}} are in the [[Open Interval|open interval]] {{open-open|1,∞}} with {{math|1/''p'' + 1/''q'' {{=}} 1}}.

===Counting measure===
For the <math>n</math>-dimensional [[Euclidean space]], when the [[Set (mathematics)|set]] <math>S</math> is <math>\{1,\dots,n\}</math> with the [[counting measure]], we have

:<math>\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\frac{1}{p}} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\frac{1}{q}}
\text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.</math>
Often the following practical form of this is used, for any <math>(r,s)\in\mathbb{R}_+</math>:
:<math>\biggl(\sum_{k=1}^n |x_k|^r\,|y_k|^s \biggr)^{r+s}\le \biggl( \sum_{k=1}^n |x_k|^{r+s} \biggr)^{r} \biggl( \sum_{k=1}^n |y_k|^{r+s} \biggr)^{s}.</math>
For more than two sums, the following generalisation ({{harvtxt|Chen|2015}}) holds, with real positive exponents   <math> \lambda_i </math> and <math> \lambda_a + \lambda_b+ \cdots +\lambda_z =1</math>:
:<math>\sum_{k=1}^n |a_k|^{\lambda_a}\,|b_k|^{\lambda_b} \cdots |z_k|^{\lambda_z} \le  \biggl(\sum_{k=1}^n |a_k|\biggr)^{\lambda_a} \biggl(\sum_{k=1}^n |b_k|\biggr)^{\lambda_b} \cdots \biggl(\sum_{k=1}^n |z_k|\biggr)^{\lambda_z} .
</math>
Equality holds iff  <math> |a_1|: |a_2|: \cdots : |a_n| =|b_1|: |b_2|: \cdots : |b_n| = \cdots = |z_1|: |z_2|: \cdots : |z_n| </math>.

If <math>S=\N</math> with the counting measure, then we get Hölder's inequality for [[sequence space]]s:

:<math>\sum_{k=1}^{\infty} |x_k\,y_k| \le \biggl( \sum_{k=1}^{\infty} |x_k|^p \biggr)^{\frac{1}{p}} \left( \sum_{k=1}^{\infty} |y_k|^q \right)^{\frac{1}{q}}
\text{ for all }(x_k)_{k\in\mathbb N}, (y_k)_{k\in\mathbb N}\in\mathbb{R}^{\mathbb N}\text{ or }\mathbb{C}^{\mathbb N}.</math>

===Lebesgue measure===
If <math>S</math> is a measurable subset of <math>\R^n</math> with the [[Lebesgue measure]], and <math>f</math> and <math>g</math> are measurable real- or complex-valued functions on <math>S</math>, then Hölder's inequality is

:<math>\int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x\biggr)^{\frac{1}{p}} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\frac{1}{q}}.</math>

===Probability measure===
For the [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P}),</math> let <math>\mathbb{E}</math> denote the [[expected value|expectation operator]]. For real- or complex-valued [[random variable]]s <math>X</math> and <math>Y</math> on <math>\Omega,</math> Hölder's inequality reads

:<math>\mathbb{E}[|XY|] \leqslant \left (\mathbb{E}\bigl[ |X|^p\bigr]\right)^{\frac{1}{p}} \left(\mathbb{E}\bigl[|Y|^q\bigr]\right)^{\frac{1}{q}}.</math>

Let <math>1 < r < s < \infty</math>  and define <math>p = \tfrac{s}{r}.</math> Then <math>q = \tfrac{p}{p-1}</math> is the Hölder conjugate of <math>p.</math> Applying Hölder's inequality to the random variables <math>|X|^r</math> and <math>1_{\Omega}</math> we obtain

:<math>\mathbb{E}\bigl[|X|^r\bigr]\leqslant \left(\mathbb{E}\bigl[|X|^s\bigr]\right)^{\frac{r}{s}}.</math>

In particular, if the {{mvar|s}}<sup>th</sup> absolute [[moment (mathematics)|moment]] is finite, then the {{mvar|r}}<sup>&nbsp;th</sup> absolute moment is finite, too. (This also follows from [[Jensen's inequality]].)

===Product measure===
For two [[σ-finite measure]] spaces {{math|(''S''<sub>1</sub>, Σ<sub>1</sub>, ''μ''<sub>1</sub>)}} and {{math|(''S''<sub>2</sub>, Σ<sub>2</sub>, ''μ''<sub>2</sub>)}} define the [[product measure space]] by

:<math>S=S_1\times S_2,\quad \Sigma=\Sigma_1\otimes\Sigma_2,\quad \mu=\mu_1\otimes\mu_2,</math>

where {{mvar|S}} is the [[Cartesian product]] of {{math|''S''<sub>1</sub>}} and {{math|''S''<sub>2</sub>}}, the {{nowrap|[[σ-algebra]] {{math|Σ}}}} arises as [[product σ-algebra]] of {{math|Σ<sub>1</sub>}} and {{math|Σ<sub>2</sub>}}, and {{mvar|μ}} denotes the [[product measure]] of {{math|''μ''<sub>1</sub>}} and {{math|''μ''<sub>2</sub>}}. Then [[Fubini's theorem#Tonelli.27s theorem|Tonelli's theorem]] allows us to rewrite Hölder's inequality using [[Iterated integral|iterated integrals]]: If&nbsp;{{mvar|f}} and {{mvar|g}} are {{nowrap|{{math|Σ}}-measurable}} real- or complex-valued functions on the Cartesian product&nbsp;{{mvar|S}}, then

:<math>\int_{S_1}\int_{S_2}|f(x,y)\,g(x,y)|\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x) \le\left(\int_{S_1}\int_{S_2}|f(x,y)|^p\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_{S_1}\int_{S_2}|g(x,y)|^q\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\right)^{\frac{1}{q}}.</math>

This can be generalized to more than two {{nowrap|σ-finite}} measure spaces.

===Vector-valued functions===
Let {{math|(''S'', Σ, ''μ'')}} denote a {{nowrap|σ-finite}} measure space and suppose that {{math|''f'' {{=}} (''f''<sub>1</sub>, ..., ''f<sub>n</sub>'')}} and {{math|''g'' {{=}} (''g''<sub>1</sub>, ..., ''g<sub>n</sub>'')}} are {{math|Σ}}-measurable functions on {{mvar|S}}, taking values in the {{mvar|n}}-dimensional real- or complex Euclidean space. By taking the product with the counting measure on {{math|{{mset|1, ..., ''n''}}}}, we can rewrite the above product measure version of Hölder's inequality in the form

:<math> \int_S \sum_{k=1}^n|f_k(x)\,g_k(x)|\,\mu(\mathrm{d}x) \le \left(\int_S\sum_{k=1}^n|f_k(x)|^p\,\mu(\mathrm{d}x)\right)^{\frac{1}{p}}\left(\int_S\sum_{k=1}^n|g_k(x)|^q\,\mu(\mathrm{d}x)\right)^{\frac{1}{q}}.</math>

If the two integrals on the right-hand side are finite, then equality holds if and only if there exist real numbers {{math|''α'', ''β'' ≥ 0}}, not both of them zero, such that

:<math>\alpha \left (|f_1(x)|^p,\ldots,|f_n(x)|^p \right )= \beta \left (|g_1(x)|^q,\ldots,|g_n(x)|^q \right ),</math>

for {{mvar|μ}}-almost all {{mvar|x}} in {{mvar|S}}.

This finite-dimensional version generalizes to functions {{mvar|f}} and {{mvar|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]].

{{Math proof|drop=hidden|proof=
If {{math|1={{norm|''f''{{hair space}}}}<sub>''p''</sub> = 0}}, then {{mvar|f}} is zero {{mvar|μ}}-almost everywhere, and the product {{math|''fg''}} is zero {{mvar|μ}}-almost everywhere, hence the left-hand side of Hölder's inequality is zero. 
The same is true if {{math|1={{norm|''g''}}<sub>''q''</sub> = 0}}. 
Therefore, we may assume {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub> > 0}} and {{math|{{norm|''g''}}<sub>''q''</sub> > 0}} in the following. 

If {{math|1={{norm|''f''{{hair space}}}}<sub>''p''</sub> = ∞}} or {{math|1={{norm|''g''}}<sub>''q''</sub> = ∞}}, then the right-hand side of Hölder's inequality is infinite. 
Therefore, we may assume that {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} and {{math|{{norm|''g''}}<sub>''q''</sub>}} are in {{open-open|0, ∞}}. 

If {{math|1=''p'' = ∞}} and {{math|1=''q'' = 1}}, then {{math|{{abs|''fg''}} ≤ {{norm|''f''{{hair space}}}}<sub>∞</sub> {{abs|''g''}}}} almost everywhere and Hölder's inequality follows from the monotonicity of the Lebesgue integral. Similarly for {{math|1=''p'' = 1}} and {{math|1=''q'' = ∞}}. 
Therefore, we may assume {{math|''p'', ''q'' ∈}} {{open-open|1,∞}}.

Dividing {{mvar|f}} and {{mvar|g}} by {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} and {{math|{{norm|''g''}}<sub>''q''</sub>}}, respectively, we can assume that

:<math>\|f\|_p = \|g\|_q = 1.</math>

We now use [[Young's inequality for products]], which states that whenever <math>p,q</math> are in {{math|(1,∞)}} with <math> \frac{1}{p} + \frac{1}{q} = 1</math>
:<math>a b \le \frac{a^p}p + \frac{b^q}q</math>

for all nonnegative {{mvar|a}} and {{mvar|b}}, where equality is achieved if and only if {{math|1=''a<sup>p</sup>'' = ''b<sup>q</sup>''}}. Hence

:<math>|f(s)g(s)| \le \frac{|f(s)|^p}p + \frac{|g(s)|^q}q,\qquad s\in S.</math>

Integrating both sides gives

:<math>\|fg\|_1 \le \frac{\|f\|_p^p}{p} + \frac{\|g\|_q^q}{q} = \frac{1}{p} + \frac{1}{q} = 1,</math>

which proves the claim.

Under the assumptions {{math|''p'' ∈}} {{open-open|1, ∞}} and {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub> {{=}} {{norm|''g''}}<sub>''q''</sub>}}, equality holds if and only if {{math|{{abs|''f''{{hair space}}}}<sup>''p''</sup> {{=}} {{abs|''g''}}<sup>''q''</sup>}} almost everywhere. 
More generally, if {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub>}} and {{math|{{norm|''g''}}<sub>''q''</sub>}} are in {{open-open|0, ∞}}, then Hölder's inequality becomes an equality if and only if there exist real numbers {{math|''α'', ''β'' > 0}}, namely

:<math>\alpha=\|g\|_q^q, \qquad \beta=\|f\|_p^p,</math>

such that

:<math>\alpha |f|^p = \beta |g|^q</math>&nbsp;&nbsp;&nbsp;''μ''-almost everywhere&nbsp;&nbsp;&nbsp;(*).

The case {{math|1={{norm|''f''{{hair space}}}}<sub>''p''</sub> = 0}} corresponds to {{math|1=''β'' = 0}} in (*). The case {{math|1={{norm|''g''}}<sub>''q''</sub> = 0}} corresponds to {{math|1=''α'' = 0}} in (*).
}}

Alternative proof using Jensen's inequality:

{{Math proof|drop=hidden|proof=
The function <math>x \mapsto x^p</math> on {{open-open|0,∞}} is convex because <math>p\geq 1</math>, so by Jensen's inequality, 

:<math>\int h\mathrm{d}\nu\leq\left(\int h^p\mathrm{d}\nu \right )^{\frac{1}{p}}</math> 

where {{mvar|ν}} is any probability distribution and {{mvar|''h''}} any {{mvar|ν}}-measurable function. Let {{mvar|μ}} be any measure, and {{mvar|ν}} the distribution whose density w.r.t. {{mvar|μ}} is proportional to <math>g^q</math>, i.e.

:<math>\mathrm{d}\nu = \frac{g^q}{\int g^q\,\mathrm{d}\mu}\mathrm{d}\mu</math>

Hence we have, using <math>\frac{1}{p}+\frac{1}{q}=1</math>, hence <math>p(1-q)+q=0</math>, and letting <math>h=fg^{1-q}</math>,

:<math> \begin{align}\int fg\,\mathrm{d}\mu = & \left (\int g^q\,\mathrm{d}\mu \right )\int \underbrace{fg^{1-q}}_h\underbrace{\frac{g^q}{\int g^q\,\mathrm{d}\mu}\mathrm{d}\mu}_{\mathrm{d}\nu}\\
 \leq & \left (\int g^q\mathrm{d}\mu \right ) \left (\int \underbrace{f^pg^{p(1-q)}}_{h^p}\underbrace{\frac{g^q}{\int g^q\,\mathrm{d}\mu}\,\mathrm{d}\mu}_{\mathrm{d}\nu} \right )^{\frac{1}{p}}\\
 = & \left (\int g^q\,\mathrm{d}\mu \right ) \left (\int \frac{f^p}{\int g^q\,\mathrm{d}\mu}\,\mathrm{d}\mu \right )^{\frac{1}{p}} .
\end{align}</math>

Finally, we get

:<math>\int fg\,\mathrm{d}\mu \leq \left(\int f^p\,\mathrm{d}\mu \right )^{\frac{1}{p}} \left(\int g^q\,\mathrm{d}\mu \right )^{\frac{1}{q}}</math>

This assumes that {{math|''f'', ''g''}} are real and non-negative, but the extension to complex functions is straightforward (use the modulus of {{math|''f'', ''g''}}). 
It also assumes that <math>\|f\|_p,\|g\|_q</math> are neither null nor infinity, and that <math>p,q > 1</math>: all these assumptions can also be lifted as in the proof above.
}}

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.

{{Math proof|drop=hidden|proof=
As in the previous proof, it suffices to prove
:<math>\int_X |h|\mathrm{d}\nu\leq\left(\int_X |h|^p\mathrm{d}\nu \right )^{\frac{1}{p}}</math> 
where <math> \nu(X)=1 </math> and  <math>h</math> is <math>\nu</math>-measurable (real or complex) function on <math> X </math>. To prove this, we must bound <math>|h| </math> by  <math> |h|^p </math>. There is no constant  <math> C </math> that will make <math> |h(x)| ~\leq~ C|h(x)|^p </math> for all  <math> x > 0 </math>. Hence, we seek an inequality of the form
:<math> |h(x)| ~\leq~ a'|h(x)|^p + b', \quad for \quad all \quad x>0 </math>
for suitable choices of  <math> a'  </math> and  <math> b' </math>.

We wish to obtain  <math> A:=\|f\|_p  </math> on the right-hand side after integrating this inequality. By trial and error, we see that the inequality we wish should have the form
:<math> |h(x)| ~\leq~ aA^{1-p}|h(x)|^p + bA, \quad for \quad all \quad x>0, </math>
where <math> a, b </math> are non-negative and <math> a+b=1 </math>. Indeed, the integral of the right-hand side is precisely <math> A </math>. So, it remains to prove that such an inequality does hold with the right choice of <math>a,b.</math>

The inequality we seek would follow from: 
:<math> \tfrac{y}{A} ~\leq~ a(\tfrac{y}{a})^p + b, \quad for \quad all \quad y>0, </math>
which, in turn, is equivalent to
:<math> (*) \quad z ~\leq~ az^p + b, \quad for \quad all \quad z>0. </math>

It turns out there is one and only one choice of <math> a, b </math>, subject to <math> a+b=1 </math>, that makes this true: <math> a=\tfrac{1}{p}</math> and, necessarily, <math> b=1-\tfrac{1}{p}</math>. (This is where Hölder conjugate exponent is born!) This completes the proof of the inequality at the first paragraph of this proof. Proof of Hölder's inequality follows from this as in the previous proof. Alternatively, we can deduce Young's inequality and then resort to the first proof given above. Young's inequality follows from the inequality (*) above by choosing <math> z=\tfrac{a}{b^{q-1}}</math> and multiplying both sides by <math> b^{q} </math>.
}}

== Extremal equality ==

=== Statement ===
Assume that {{math|1 ≤ ''p'' < ∞}} and let {{mvar|q}} denote the Hölder conjugate. Then for every {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}},

:<math>\|f\|_p = \max \left \{ \left| \int_S f g \, \mathrm{d}\mu \right | : g\in L^q(\mu), \|g\|_q \le 1 \right\},</math>

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

:<math>\|f\|_\infty = \sup \left\{ \left| \int_S f g \,\mathrm{d}\mu \right| : g\in L^1(\mu), \|g\|_1 \le 1 \right \}.</math>
Proof of the extremal equality:

{{Math proof|drop=hidden|proof=
By Hölder's inequality, the integrals are well defined and, for {{math|1 ≤ ''p'' ≤ ∞}},

:<math>\left |\int_S fg\,\mathrm{d}\mu\right|\le\int_S|fg|\,\mathrm{d}\mu\le\|f\|_p,</math>

hence the left-hand side is always bounded above by the right-hand side.

Conversely, for {{math|1 ≤ ''p'' ≤ ∞}}, observe first that the statement is obvious when {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub> {{=}} 0}}. Therefore, we assume {{math|{{norm|''f''{{hair space}}}}<sub>''p''</sub> > 0}} in the following.

If {{math|1 ≤ ''p'' < ∞}}, define {{mvar|g}} on {{mvar|S}} by

:<math>g(x) = \begin{cases}\|f\|_p^{1-p} \, |f(x)|^p / f(x)&\text{if }f(x)\not=0,\\ 0&\text{otherwise.}\end{cases}</math>

By checking the cases {{math|''p'' {{=}} 1}} and {{math|1 < ''p'' < ∞}} separately, we see that {{math|{{norm|''g''}}<sub>''q''</sub> {{=}} 1}} and

:<math>\int_S f g \, \mathrm{d}\mu = \|f\|_p.</math>

It remains to consider the case {{math|''p'' {{=}} ∞}}. For {{math|''ε'' ∈}} {{open-open|0, 1}} define

:<math>A=\left \{x\in S:|f(x)|>(1-\varepsilon)\|f\|_\infty\right\}.</math>

Since {{mvar|f}} is measurable, {{math|''A'' ∈ Σ}}. By the definition of {{math|{{norm|''f''{{hair space}}}}<sub>∞</sub>}} as the [[essential supremum]] of {{mvar|f}} and the assumption {{math|{{norm|''f''{{hair space}}}}<sub>∞</sub> > 0}}, we have {{math|''μ''(''A'') > 0}}. Using the additional assumption on the {{nowrap|σ-field}} {{math|Σ}} if necessary, there exists a subset {{math|''B'' ∈ Σ}} of {{mvar|A}} with {{math|0 < ''μ''(''B'') < ∞}}. Define {{mvar|g}} on {{mvar|S}} by

:<math>g(x)=\begin{cases}\frac{1-\varepsilon}{\mu(B)}\frac{\|f\|_\infty}{f(x)}&\text{if }x\in B,\\0&\text{otherwise.}\end{cases}</math>

Then {{mvar|g}} is well-defined, measurable and {{math|{{abs|''g''(''x'')}} ≤ 1/''μ''(''B'')}} for {{math|''x'' ∈ ''B''}}, hence {{math|{{norm|''g''}}<sub>1</sub> ≤ 1}}. Furthermore,

:<math>\left |\int_S fg\,\mathrm{d}\mu\right| = \int_B\frac{1-\varepsilon}{\mu(B)}\|f\|_\infty\,\mathrm{d}\mu = (1-\varepsilon)\|f\|_\infty.</math>
}}

=== Remarks and examples ===

* The equality for <math>p = \infty</math> fails whenever there exists a set <math>A</math> of infinite measure in the <math>\sigma</math>-field <math>\Sigma</math> with that has no subset <math>B \in \Sigma</math> that satisfies: <math>0 < \mu(B) < \infty.</math> (the simplest example is the <math>\sigma</math>-field  <math>\Sigma</math> containing just the empty set and <math>S,</math> and the measure <math>\mu</math> with <math>\mu(S) = \infty.</math>) Then the [[indicator function]] <math>1_A</math> satisfies <math>\|1_A\|_{\infty} = 1,</math> but every <math>g \in L^1 (\mu)</math> has to be <math>\mu</math>-almost everywhere constant on <math>A,</math> because it is <math>\Sigma</math>-measurable, and this constant has to be zero, because <math>g</math> is <math>\mu</math>-integrable. Therefore, the above supremum for the indicator function <math>1_A</math> is zero and the extremal equality fails.
* For <math>p = \infty,</math> the supremum is in general not attained. As an example, let <math>S = \mathbb{N}, \Sigma = \mathcal{P}(\mathbb{N})</math> and <math>\mu</math> the counting measure. Define:

::<math>\begin{cases} f: \mathbb{N} \to \mathbb{R} \\ f(n) = \frac{n-1}{n} \end{cases}</math>

:Then <math>\|f\|_{\infty} = 1.</math> For <math>g \in L^1 (\mu, \mathbb{N})</math> with <math>0 < \|g\|_1 \leqslant 1,</math> let <math>m</math> denote the smallest natural number with <math>g(m) \neq 0.</math> Then

::<math>\left |\int_S fg\,\mathrm{d}\mu\right| \leqslant \frac{m-1}{m}|g(m)|+\sum_{n=m+1}^\infty|g(n)| = \|g\|_1-\frac{|g(m)|}m<1.</math>

=== Applications ===
*The extremal equality is one of the ways for proving the triangle inequality {{math|{{norm|''f''<sub>1</sub> + ''f''<sub>2</sub>}}<sub>''p''</sub> ≤ {{norm|''f''<sub>1</sub>}}<sub>''p''</sub> + {{norm|''f''<sub>2</sub>}}<sub>''p''</sub>}} for all {{math|''f''<sub>1</sub>}} and {{math|''f''<sub>2</sub>}} in {{math|''L<sup>p</sup>''(''μ'')}}, see [[Minkowski inequality]].
*Hölder's inequality implies that every {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}} defines a bounded (or continuous) linear functional {{math|''κ<sub>f</sub>''}} on {{math|''L<sup>q</sup>''(''μ'')}} by the formula
::<math>\kappa_f(g) = \int_S f g \, \mathrm{d}\mu,\qquad g\in L^q(\mu).</math>
:The extremal equality (when true) shows that the norm of this functional {{math|''κ<sub>f</sub>''}} as element of the [[continuous dual space]] {{math|''L<sup>q</sup>''(''μ'')<sup>*</sup>}} coincides with the norm of {{mvar|f}} in {{math|''L<sup>p</sup>''(''μ'')}} (see also the {{nowrap|[[L^p-space|{{math|''L<sup>p</sup>''}}-space]]}} article).

==Generalization with more than two functions==

=== Statement ===
Assume that {{math|''r'' ∈}} {{open-closed|0, ∞}} and {{math|''p''<sub>1</sub>, ..., ''p<sub>n</sub>'' ∈ }} {{open-closed|0, ∞}} such that

:<math>\sum_{k=1}^n \frac1{p_k} = \frac1r</math>

where 1/∞ is interpreted as 0 in this equation.  Then for all measurable real or complex-valued functions {{math|''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>}} defined on {{mvar|S}},

:<math>\left\|\prod_{k=1}^n f_k\right\|_r \le \prod_{k=1}^n \left\|f_k\right\|_{p_k}</math>

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 <math>f_k \in L^{p_k}(\mu)</math> for all <math>k \in \{ 1, \ldots, n \}</math> then <math>\prod_{k=1}^n f_k \in L^r(\mu).</math>

'''Note:''' For <math>r \in (0, 1),</math> contrary to the notation, {{math|{{norm|.}}<sub>''r''</sub>}} is in general not a norm because it doesn't satisfy the [[triangle inequality]].

Proof of the generalization:
{{Math proof|drop=hidden|proof=
We use Hölder's inequality and [[mathematical induction]]. If <math>n = 1</math> then the result is immediate. Let us now pass from <math>n - 1</math> to <math>n.</math> Without loss of generality assume that <math>p_1 \leq \cdots \leq p_n.</math>

'''Case 1:''' If <math>p_n = \infty</math> then

:<math>\sum_{k=1}^{n-1} \frac1{p_k} = \frac1r.</math>

Pulling out the essential supremum of {{math|{{abs|''f<sub>n</sub>''}}}} and using the induction hypothesis, we get
:<math>\begin{align}
\left\|f_1 \cdots f_n \right\|_r 
&\le \left\|f_1 \cdots f_{n-1}\right\|_r \left\|f_n\right\|_{\infty}\\ 
&\le \left\|f_1\right\|_{p_1} \cdots \left\|f_{n-1}\right\|_{p_{n-1}} \left\|f_n\right\|_{\infty}.
\end{align}</math>

'''Case 2:''' If <math>p_n < \infty</math> then necessarily <math>r < \infty</math> as well, and then

:<math>p := \frac{p_n}{p_n-r}, \qquad q := \frac{p_n}r</math>

are Hölder conjugates in {{open-open|1, ∞}}. Application of Hölder's inequality gives

:<math>\left \||f_1 \cdots f_{n-1}|^r\,|f_n|^r\right \|_1 \le \left \||f_1 \cdots f_{n-1}|^r\right\|_p\,\left \||f_n|^r\right \|_q.</math>

Raising to the power <math>1/r</math> and rewriting,

:<math>\|f_1 \cdots f_n\|_r \le \|f_1 \cdots f_{n-1}\|_{pr} \|f_n\|_{qr}.</math>

Since <math>q r = p_n</math> and

:<math>\sum_{k=1}^{n-1} \frac1{p_k} = \frac1r-\frac1{p_n} = \frac{p_n-r}{rp_n} = \frac1{pr},</math>

the claimed inequality now follows by using the induction hypothesis.
}}

===Interpolation===
Let {{math|''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub> ∈}}  {{open-closed|0, ∞}} and let {{math|''θ''<sub>1</sub>, ..., ''θ''<sub>''n''</sub> ∈ (0, 1)}} denote weights with {{math|''θ''<sub>1</sub> + ... + ''θ<sub>n</sub>'' {{=}} 1}}. Define <math>p</math> as the weighted [[harmonic mean]], that is,

:<math> \frac1p = \sum_{k=1}^n \frac{\theta_k}{p_k}.</math>

Given measurable real- or complex-valued functions <math>f_k</math> on {{mvar|S}}, then the above generalization of Hölder's inequality gives
:<math>\left\| |f_1|^{\theta_1}\cdots |f_n|^{\theta_n}\right\|_p \le \left\||f_1|^{\theta_1}\right\|_{\frac{p_1}{\theta_1}}\cdots \left\| |f_n|^{\theta_n}\right\|_{\frac{p_n}{\theta_n}} = \|f_1\|_{p_1}^{\theta_1}\cdots \|f_n\|_{p_n}^{\theta_n}.</math>

In particular, taking <math>f_1 = \cdots = f_n=:f</math> gives
:<math>\|f\|_p \leqslant \prod_{k=1}^n \|f\|_{p_k}^{\theta_k}.</math>

Specifying further {{math|''θ''<sub>1</sub> {{=}} ''θ''}} and {{math|''θ''<sub>2</sub> {{=}} 1-''θ''}}, in the case <math>n = 2,</math> we obtain the [[Riesz-Thorin theorem|interpolation]] result 

{{Math theorem
| name = Littlewood's inequality
| note = 
| math_statement = For <math>\theta\in(0,1)</math> and <math>\frac {1}{p_\theta} = \frac{\theta}{p_1} + \frac {1-\theta}{p_0}</math>, 

<math display="block">\|f\|_{p_\theta} \leqslant \|f\|_{p_1}^\theta \cdot \|f\|_{p_0}^{1-\theta},</math>
}}

An application of Hölder gives 

{{Math theorem
| name = Lyapunov's inequality
| note = 
| math_statement = If <math>p = (1-\theta) p_0 + \theta p_1, \qquad \theta \in (0, 1),</math> then

<math display="block">\left\| |f_0|^{\frac{p_0(1-\theta)}{p}} \cdot |f_1|^{\frac{p_1 \theta}{p}}\right\|_p^p \le \|f_0\|_{p_0}^{p_0(1-\theta)} \|f_1\|_{p_1}^{p_1\theta}</math>

and in particular

<math display="block">\|f\|_p^p \leqslant \|f\|_{p_0}^{p_0(1-\theta)} \cdot \|f\|_{p_1}^{p_1\theta}.</math>
}}

Both Littlewood and Lyapunov imply that if <math>f \in L^{p_0}\cap L^{p_1}</math> then <math>f \in L^p</math> for all <math>p_0 < p < p_1.</math><ref>{{Cite book |last=Wojtaszczyk |first=P. |url=https://www.cambridge.org/core/books/banach-spaces-for-analysts/7647AD5BB01A5ABBC96789F74DDE8A2F |title=Banach Spaces for Analysts |date=1991 |publisher=Cambridge University Press |isbn=978-0-521-56675-9 |series=Cambridge Studies in Advanced Mathematics |location=Cambridge}}</ref>

== Reverse Hölder inequalities ==

=== Two functions ===
Assume that {{math|''p'' ∈ (1, ∞)}} and that the measure space {{math|(''S'', Σ, ''μ'')}} satisfies {{math|''μ''(''S'') > 0}}. Then for all measurable real- or complex-valued functions {{mvar|f}} and {{mvar|g}} on {{mvar|S}} such that {{math|''g''(''s'') ≠ 0}} for {{nowrap|{{mvar|μ}}-almost}} all {{math|''s'' ∈ ''S''}},

:<math>\|fg\|_1\geqslant \|f\|_{\frac{1}{p}}\,\|g\|_{\frac{-1}{p-1}}.</math>

If

:<math>\|fg\|_1 < \infty \quad \text{and} \quad \|g\|_{\frac{-1}{p-1}} > 0, </math>

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

:<math>\exists \alpha \geqslant 0 \quad |f| = \alpha|g|^{\frac{-p}{p-1}} \qquad \mu\text{-almost everywhere}.</math>

'''Note:''' The expressions:

<math> \|f\|_{\frac{1}{p}}</math> and <math>\|g\|_{\frac{-1}{p-1}},</math>

are not norms, they are just compact notations for

:<math>\left (\int_S|f|^{\frac{1}{p}}\,\mathrm{d}\mu\right)^{p} \quad \text{and} \quad \left (\int_S|g|^{\frac{-1}{p-1}}\,\mathrm{d}\mu\right)^{-(p-1)}.</math>

{{hidden
|Proof of the reverse Hölder inequality (hidden, click show to reveal.)
|2= 

Note that {{mvar|p}} and

:<math>q:=\frac{p}{p-1}\in(1,\infty)</math>

are Hölder conjugates. 
Application of Hölder's inequality gives

:<math>\begin{align}
\left \||f|^{\frac{1}{p}}\right \|_1 &= \left \||fg|^{\frac{1}{p}}\,|g|^{-\frac{1}{p}}\right \|_1\\
&\leqslant \left \| |fg|^{\frac{1}{p}} \right \|_p \left \| |g|^{-\frac{1}{p}}\right \|_q \\
&=\|fg\|_1^{\frac{1}{p}}\left \||g|^{\frac{-1}{p-1}}\right \|_1^{\frac{p-1}{p}}
\end{align}</math>

Raising to the power {{mvar|p}} gives us:

:<math>\left \||f|^{\frac{1}{p}}\right \|_1^p \leqslant \|fg\|_1  \left \||g|^{\frac{-1}{p-1}}\right \|_1^{p-1}.</math>

Therefore:

:<math>\left \||f|^{\frac{1}{p}}\right \|_1^p   \left \||g|^{\frac{-1}{p-1}}\right \|_1^{-(p-1)} \leqslant \|fg\|_1 .</math>

Now we just need to recall our notation.

Since {{mvar|g}} is not almost everywhere equal to the zero function, we can have equality if and only if there exists a constant {{math|''α'' ≥ 0}} such that {{math|{{abs|''fg''}} {{=}} ''α''{{hair space}}{{abs|''g''}}<sup>−''q''/''p''</sup>}} almost everywhere. Solving for the absolute value of {{mvar|f}} gives the claim.
}}

=== 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 <math>p_1,..., p_{m-1} < 0</math> and <math>p_m \in \mathbb{R}</math> be such that <math>\sum_{k=1}^{m} \frac{1}{p_k} = 1</math> (hence <math>0 < p_m < 1</math>). Let <math>f_k</math> be measurable functions for <math>k = 1,...,m</math>. Then

:<math>\left\|\prod_{k=1}^m f_k\right\|_1 \ge \prod_{k=1}^m \left\|f_k\right\|_{p_k}.</math>

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

== Symmetric forms of Hölder inequality == 

It was observed by [[János_Aczél_(mathematician)|Aczél]] and [[Edwin_F._Beckenbach|Beckenbach]]<ref>{{cite book |last1=Beckenbach |first1=E. F. |title=General inequalities 2 |series=International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d'Analyse Numérique |date=1980 |volume=47 |publisher=Birkhäuser Basel |isbn=978-3-7643-1056-1 |pages=145–150 |doi=10.1007/978-3-0348-6324-7 |url=https://doi.org/10.1007/978-3-0348-6324-7}}</ref> that Hölder's inequality can be put in a more symmetric form, at the price of introducing an extra vector (or function): 

Let <math>f = (f(1),\dots,f(m)) , g = (g(1),\dots, g(m)), h = (h(1),\dots,h(m))</math> be vectors with positive entries and such that <math>f(i) g(i) h(i) = 1</math> for all <math>i</math>. If <math>p,q,r</math> are nonzero real numbers such that <math>\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=0</math>, then:
* <math>\|f\|_p \|g\|_q \|h\|_r \ge 1</math> if all but one of <math>p,q,r</math> are positive;
* <math>\|f\|_p \|g\|_q \|h\|_r \le 1</math> if all but one of <math>p,q,r</math> 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 <math>f_1, \dots, f_n</math> be <math>n</math> vectors in <math>\mathbb{R}^m</math> with positive entries and such that <math>f_1(i) \dots f_n(i) = 1</math> for all <math>i</math>. If <math>p_1,\dots,p_n</math> are nonzero real numbers such that <math>\frac{1}{p_1}+\dots+\frac{1}{p_n}=0</math>, then:
* <math>\|f_1\|_{p_1} \dots \|f_n\|_{p_n} \ge 1</math> if all but one of the numbers <math>p_i</math> are positive;
* <math>\|f_1\|_{p_1} \dots \|f_n\|_{p_n} \le 1</math> if all but one of the numbers <math>p_i</math> are negative.

As in the standard Hölder inequalities, there are corresponding statements for infinite sums and integrals.

== Conditional Hölder inequality ==
Let {{math|(Ω, {{mathcal|F}}, <math>\mathbb{P}</math>)}} be a probability space, {{math|{{mathcal|G}} ⊂ {{mathcal|F}}}} a {{nowrap|sub-[[σ-algebra]]}}, and {{math|''p'', ''q'' ∈}} {{open-open|1, ∞}} Hölder conjugates, meaning that {{math|1/''p'' + 1/''q'' {{=}} 1}}. Then for all real- or complex-valued random variables {{mvar|X}} and {{mvar|Y}} on&nbsp;{{math|Ω}},

:<math>\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr] \le \bigl(\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{p}} \,\bigl(\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{q}}

\qquad\mathbb{P}\text{-almost surely.}</math>

'''Remarks:'''
* If a non-negative random variable {{mvar|Z}} has infinite [[expected value]], then its [[conditional expectation]] is defined by
::<math>\mathbb{E}[Z|\mathcal{G}] = \sup_{n\in\mathbb{N}}\,\mathbb{E}[\min\{Z,n\}|\mathcal{G}]\quad\text{a.s.}</math>

* On the right-hand side of the conditional Hölder inequality, 0 times ∞ as well as ∞ times 0 means&nbsp;0. Multiplying {{math|''a'' > 0}} with ∞ gives&nbsp;∞.

Proof of the conditional Hölder inequality:
{{Math proof|drop=hidden|proof=
Define the random variables

:<math>U=\bigl(\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{p}},\qquad V=\bigl(\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]\bigr)^{\frac{1}{q}}</math>

and note that they are measurable with respect to the {{nowrap|sub-σ-algebra}}. Since

:<math>\mathbb{E}\bigl[|X|^p1_{\{U=0\}}\bigr] = \mathbb{E}\bigl[1_{\{U=0\}}\underbrace{\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]}_{=\,U^p}\bigr]=0,</math>

it follows that {{math|{{abs|''X''}} {{=}} 0}} a.s. on the set {{math|{{mset|''U'' {{=}} 0}}}}. Similarly, {{math|{{abs|''Y''}} {{=}} 0}} a.s. on the set {{math|{{mset|''V'' {{=}} 0}}}}, hence

:<math>\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]=0\qquad\text{a.s. on }\{U=0\}\cup\{V=0\}</math>

and the conditional Hölder inequality holds on this set. On the set

:<math>\{U=\infty, V>0\}\cup\{U>0, V=\infty\}</math>

the right-hand side is infinite and the conditional Hölder inequality holds, too. Dividing by the right-hand side, it therefore remains to show that

:<math>\frac{\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]}{UV}\le1
\qquad\text{a.s. on the set }H:=\{0<U<\infty,\,0<V<\infty\}.</math>

This is done by verifying that the inequality holds after integration over an arbitrary

:<math>G\in\mathcal{G},\quad G\subset H.</math>

Using the measurability of {{mvar|U, V, 1<sub>''G''</sub>}} with respect to the {{nowrap|sub-σ-algebra}}, the rules for conditional expectations, Hölder's inequality and {{math|1/''p'' + 1/''q'' {{=}} 1}}, we see that

:<math>\begin{align}
\mathbb{E}\biggl[\frac{\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]}{UV}1_G\biggr]
&=\mathbb{E}\biggl[\mathbb{E}\biggl[\frac{|XY|}{UV}1_G\bigg|\,\mathcal{G}\biggr]\biggr]\\
&=\mathbb{E}\biggl[\frac{|X|}{U}1_G\cdot\frac{|Y|}{V}1_G\biggr]\\
&\le\biggl(\mathbb{E}\biggl[\frac{|X|^p}{U^p}1_G\biggr]\biggr)^{\frac{1}{p}}
\biggl(\mathbb{E}\biggl[\frac{|Y|^q}{V^q}1_G\biggr]\biggr)^{\frac{1}{q}}\\
&=\biggl(\mathbb{E}\biggl[\underbrace{\frac{\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]}{U^p}}_{=\,1\text{ a.s. on }G}1_G\biggr]\biggr)^{\frac{1}{p}}
\biggl(\mathbb{E}\biggl[\underbrace{\frac{\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]}{V^p}}_{=\,1\text{ a.s. on }G}1_G\biggr]\biggr)^{\frac{1}{q}}\\
&=\mathbb{E}\bigl[1_G\bigr].
\end{align}</math>
}}

==Hölder's inequality for increasing seminorms==
Let {{mvar|S}} be a set and let <math>F(S, \mathbb{C})</math> be the space of all complex-valued functions on {{mvar|S}}. Let {{mvar|N}} be an increasing [[seminorm]] on <math>F(S, \mathbb{C}),</math> meaning that, for all real-valued functions <math>f, g \in F(S, \mathbb{C})</math> we have the following implication (the seminorm is also allowed to attain the value ∞):

:<math> \forall s \in S \quad  f(s) \geqslant g(s) \geqslant 0  \qquad \Rightarrow \qquad  N(f) \geqslant N(g).</math>

Then:

:<math>\forall f, g \in F(S, \mathbb{C}) \qquad N(|fg|) \leqslant \bigl(N(|f|^p)\bigr)^{\frac{1}{p}} \bigl(N(|g|^q)\bigr)^{\frac{1}{q}},</math>

where the numbers <math>p</math> and <math>q</math> are Hölder conjugates.<ref>For a proof see {{harv|Trèves|1967|loc=Lemma&nbsp;20.1, pp.&nbsp;205–206}}.</ref>

'''Remark:''' If {{math|(''S'', Σ, ''μ'')}} is a [[measure space]] and <math>N(f)</math> is the upper Lebesgue integral of <math>|f|</math> then the restriction of {{mvar|N}} to all {{nowrap|{{math|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<ref>{{cite journal|last1=Nielsen |first1=Frank  |last2=Sun |first2=Ke |last3=Marchand-Maillet |first3=Stephane
|title= On Hölder projective divergences
| year = 2017
  | volume=3
| number=19
  | journal=Entropy
  | pages=122
 |doi=10.3390/e19030122 |arxiv=1701.03916 |bibcode=2017Entrp..19..122N |doi-access=free }}</ref> 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==
{{reflist}}

== References ==
* {{Citation | last1=Grinshpan | first1=A. Z. | title=Weighted inequalities and negative binomials | doi=10.1016/j.aam.2010.04.004 | year=2010 | 
journal=Advances in Applied Mathematics | volume=45 | issue=4 | pages=564–606 | doi-access=free }}
* {{citation
 | last1=Hardy
 | first1=G. H. 
 | author-link=Godfrey Harold Hardy
 | last2=Littlewood
 | first2= J. E. 
 | author2-link=John Edensor Littlewood
 | last3=Pólya
 | first3=G.
 | author3-link=George Pólya
 | title=Inequalities
 | publisher= [[Cambridge University Press]] 
 | pages=XII+314
 | year=1934
 | isbn=0-521-35880-9
 | jfm=60.0169.01 
 | zbl=0010.10703
}}.
* {{citation
 | last= Hölder
 | first=O.
 | author-link = Otto Hölder
 | title= Ueber einen Mittelwertsatz 
 | journal= Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen
 | series=Band
 | url = http://resolver.sub.uni-goettingen.de/purl?GDZPPN00252421X
 | volume = 1889
 | issue =2
 | year = 1889
 | language = de
 | pages = 38–47
 | jfm = 21.0260.07
}}. Available at [https://web.archive.org/web/20090908025223/http://www.digizeitschriften.de/index.php?id=64&L=2 Digi Zeitschriften].
* {{springer
 | last = Kuptsov
 | first = L. P. 
 | id = H/h047514
 | title = Hölder inequality
}}.
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{Citation
  | last = Rogers
  | first = L. J.
  | author-link = Leonard James Rogers
  | title = An extension of a certain theorem in inequalities
  | journal = [[Messenger of Mathematics]]
  | series = New Series
  | volume = XVII
  | issue = 10
  | pages = 145–150
  |date=February 1888
  | url = https://archive.org/details/messengermathem01unkngoog
  | archive-url = https://archive.org/stream/messengermathem01unkngoog#page/n183/mode/1up
  | archive-date = August 21, 2007
  | jfm = 20.0254.02
}}.
*{{Citation 
 | last=Roman 
 | first=Stephen
 | title=Advanced Linear Algebra 
 | edition=Third 
 | series=[[Graduate Texts in Mathematics]] 
 | publisher  = Springer 
 | date=2008
 | pages= 
 | isbn=978-0-387-72828-5 
 |author-link=Steven Roman
}}
* {{Citation
  | last = Trèves
  | first = François
  | title = Topological Vector Spaces, Distributions and Kernels
  | publisher = Academic Press
  | place = New York, London
  | series = Pure and Applied Mathematics. A Series of Monographs and Textbooks
  | volume = 25
  | year = 1967
  | mr = 0225131
  | zbl = 0171.10402
}}.
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->

==External links==
*{{Citation
|last=Chen
|first=Evan
|title=A Brief Introduction to Olympiad Inequalities
|url=http://www.mit.edu/~evanchen/handouts/Ineq/en.pdf
|year=2015
}}.
*{{Citation
 | last=Kuttler
 | first=Kenneth
 | title=An Introduction to Linear Algebra
 | publisher=Online e-book in PDF format, Brigham Young University
 | url=http://www.math.byu.edu/~klkuttle/Linearalgebra.pdf
 | year=2007
 | access-date=2008-03-26
 | archive-date=2008-08-07
 | archive-url=https://web.archive.org/web/20080807175104/http://www.math.byu.edu/~klkuttle/Linearalgebra.pdf
 | url-status=dead
 }}.
*{{Citation
|last=Lohwater
|first=Arthur
|title=Introduction to Inequalities
|format=PDF
|url=http://www.mediafire.com/file/1mw1tkgozzu
|year=1982
}}.
*Archived at [https://ghostarchive.org/varchive/youtube/20211211/kxQiKaIuyOg Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20131025042637/http://www.youtube.com/watch?v=kxQiKaIuyOg&list=PL7BBBB7E2F4C5B19F Wayback Machine]{{cbignore}}: {{Citation
 | last=Tisdell
 | first=Chris
 | title=Holder's Inequality
 | publisher=YouTube
 | url=https://www.youtube.com/watch?v=kxQiKaIuyOg
 | year=2012
}}{{cbignore}}.

{{Lp spaces}}
{{Measure theory}}
{{Functional analysis}}

{{DEFAULTSORT:Holder's inequality}}
[[Category:Inequalities]]
[[Category:Probabilistic inequalities]]
[[Category:Theorems in functional analysis]]
[[Category:Articles containing proofs]]
[[Category:Lp spaces]]