View source for Hölder's inequality

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]].

:'''Theorem (Hölder's inequality).''' 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>\|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 iff {{math|{{abs|''f''&thinsp;}}<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''&thinsp;{{!}}<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]]. Hölder's inequality holds even if {{math|{{norm|''fg''}}<sub>1</sub>}} is infinite, the right-hand side also being infinite in that case. Conversely, if {{mvar|f}}&thinsp; 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 was first found by [[Leonard James Rogers]] ({{harvtxt|Rogers|1888}}), and discovered independently by {{harvtxt|Hölder|1889}}.

==Remarks==

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

* In the definition of Hölder conjugates, {{math|1/&thinsp;∞}} means zero.
* If {{math|''p'', ''q'' ∈}} {{closed-open|1, ∞}}, then {{math|{{norm|''f''&thinsp;}}<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''&thinsp;}}<sub>∞</sub>}} stands for the [[essential supremum]] of {{math|{{abs|''f''&thinsp;}}}}, similarly for {{math|{{norm|''g''}}<sub>∞</sub>}}.
* The notation {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub>}} with {{math|1 ≤ ''p'' ≤ ∞}} is a slight abuse, because in general it is only a [[norm (mathematics)|norm]] of {{mvar|f}}&thinsp; if {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub>}} is finite and {{mvar|f}}&thinsp; 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}}&thinsp; 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}}&thinsp; 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}}&thinsp; 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''&thinsp;}}<sub>2</sub>}} and {{math|{{norm|''g''}}<sub>2</sub>}} are finite to make sure that the inner product of {{mvar|f}}&thinsp; 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''&thinsp;}}}} and {{math|{{abs|''g''}}}} in place of {{mvar|f}}&thinsp; and {{mvar|g}}.

===Generalization for probability measures===
If {{math|(''S'',&thinsp;Σ,&thinsp;''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}}&thinsp; 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-open|1,∞}} with {{math|1/''p'' + 1/''q'' {{=}} 1}}.

===Counting measure===
For the {{mvar|n}}-dimensional [[Euclidean space]], when the set {{mvar|S}} is {{math|{{mset|1, ..., ''n''}}}} 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>

If {{math|''S'' {{=}} '''N'''}} 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 {{mvar|S}} is a measurable subset of {{math|'''R'''<sup>''n''</sup>}} with the [[Lebesgue measure]], and {{mvar|f}}&thinsp; and {{mvar|g}} are measurable real- or complex-valued functions on&nbsp;{{mvar|S}}, then Hölder 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>0 < r< s</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|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 integrals: If&nbsp;{{mvar|f}}&thinsp; 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}}&thinsp; 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]].

If {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub> {{=}} 0}}, then {{mvar|f}}&thinsp; 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|{{norm|''g''}}<sub>''q''</sub> {{=}} 0}}. Therefore, we may assume {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub> > 0}} and {{math|{{norm|''g''}}<sub>''q''</sub> > 0}} in the following.

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

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

Dividing {{mvar|f}}&thinsp; and {{mvar|g}} by {{math|{{norm|''f''&thinsp;}}<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

:<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|''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,&thinsp;∞}} and {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub> {{=}} {{norm|''g''}}<sub>''q''</sub>}}, equality holds if and only if {{math|{{abs|''f''&thinsp;}}<sup>''p''</sup> {{=}} {{abs|''g''}}<sup>''q''</sup>}} almost everywhere. More generally, if {{math|{{norm|''f''&thinsp;}}<sub>''p''</sub>}} and {{math|{{norm|''g''}}<sub>''q''</sub>}} are in {{open-open|0,&thinsp;∞}}, 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|{{norm|''f''&thinsp;}}<sub>''p''</sub> {{=}} 0}} corresponds to {{math|''β'' {{=}} 0}} in (*). The case {{math|{{norm|''g''}}<sub>''q''</sub> {{=}} 0}} corresponds to {{math|''α'' {{=}} 0}} in (*).

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<div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Alternate proof using Jensen's inequality</div>
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Recall the [[Jensen's inequality]] for the convex function <math>x^p</math> (it is convex because obviously <math>p\geq1</math>):

:<math>\int hd\nu\leq\left(\int h^pd\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>d\nu = \frac{g^q}{\int g^q\,d\mu}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>\int fg\,d\mu = \left (\int g^q\,d\mu \right )\int \underbrace{fg^{1-q}}_h\underbrace{\frac{g^q}{\int g^q\,d\mu}d\mu}_{d\nu} \leq \left (\int g^qd\mu \right ) \left (\int \underbrace{f^pg^{p(1-q)}}_{h^p}\underbrace{\frac{g^q}{\int g^q\,d\mu}\,d\mu}_{d\nu} \right )^{\frac{1}{p}} = \left (\int g^q\,d\mu \right ) \left (\int \frac{f^p}{\int g^q\,d\mu}\,d\mu \right )^{\frac{1}{p}}</math>

Finally, we get

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

This assumes <math>f,g</math> real and non negative, but the extension to complex functions is straightforward (use the modulus of <math>f,g</math>). 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.
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== 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>

<div style="clear:both;width:95%;" class="NavFrame">
<div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Proof of the extremal equality</div>
<div class="NavContent" style="text-align:left;display:none;">

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''&thinsp;}}<sub>''p''</sub> {{=}} 0}}. Therefore, we assume {{math|{{norm|''f''&thinsp;}}<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,&thinsp;1}} define

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

Since {{mvar|f}}&thinsp; is measurable, {{math|''A'' ∈ Σ}}. By the definition of {{math|{{norm|''f''&thinsp;}}<sub>∞</sub>}} as the [[essential supremum]] of {{mvar|f}}&thinsp; and the assumption {{math|{{norm|''f''&thinsp;}}<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>
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=== 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>''}}&thinsp; 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>''}}&thinsp; as element of the [[continuous dual space]] {{math|''L<sup>q</sup>''(''μ'')<sup>*</sup>}} coincides with the norm of {{mvar|f}}&thinsp; in {{math|''L<sup>p</sup>''(''μ'')}} (see also the {{nowrap|[[L^p-space|{{math|''L<sup>p</sup>''}}-space]]}} article).

==Generalization of Hölder's inequality==
Assume that {{math|''r'' ∈}} {{open-closed|0,&thinsp;∞}} and {{math|''p''<sub>1</sub>, …, ''p<sub>n</sub>'' ∈ }} {{open-closed|0,&thinsp;∞}} such that

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

(where we interpret 1/∞ 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\|f_k\|_{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,

:<math>f_k\in L^{p_k}(\mu)\;\;\forall k\in\{1,\ldots,n\}\implies\prod_{k=1}^n f_k \in L^r(\mu).</math>

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

<div style="clear:both;width:95%;" class="NavFrame">
<div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Proof of the generalization</div>
<div class="NavContent" style="text-align:left;display:none;">

We use Hölder's inequality and [[mathematical induction]]. For {{math|''n'' {{=}} 1}}, the result is obvious. Let us now pass from {{math|''n'' − 1}} to {{mvar|n}}. Without loss of generality assume that {{math|''p''<sub>1</sub> ≤ … ≤ ''p<sub>n</sub>''}}.

'''Case 1:''' If {{math|''p<sub>n</sub>'' {{=}} ∞}}, 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 \|f_1\cdots f_{n-1}\|_r \|f_n\|_\infty\\ 
&\le\|f_1\|_{p_1}\cdots\|f_{n-1}\|_{p_{n-1}}\|f_n\|_\infty.
\end{align}</math>

'''Case 2:''' If {{math|''p<sub>n</sub>'' < ∞}}, then necessarily {{math|''r'' < ∞}} 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,&thinsp;∞}}. 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''}} and rewriting,

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

Since {{math|''qr'' {{=}} ''p<sub>n</sub>''}} 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.

</div>
</div>

===Interpolation===
Let {{math|''p''<sub>1</sub>, ..., ''p<sub>n</sub>'' ∈}}  {{open-closed|0,&thinsp;∞}} 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 {{mvar|p}} as the weighted [[harmonic mean]], i.e.,

:<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}}, we obtain the [[Riesz-Thorin theorem|interpolation]] result (Littlewood's inequality)

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

for <math>\theta\in(0,1)</math> and

:<math>\frac {1}{p_\theta}= \frac{\theta}{p_1}+ \frac {1-\theta}{p_0}.</math>

An application of Hölder gives Lyapunov's inequality: If

:<math>p=(1-\theta) p_0+\theta p_1, \qquad \theta\in(0,1),</math>

then

:<math>\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>\|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>

<!--P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Presse 1991, [https://books.google.com/books?id=Re4pqY72Bo8C&printsec=frontcover&dq=banach+spaces+for+analysts+wojtaszczyk&ei=HP2pSrX9H6CGygS7qfmGCg&hl=de#v=onepage&q=&f=false]</ref>-->

== Reverse Hölder inequality ==
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}}&thinsp; 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}} \quad \text{and} \quad \|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>

<div style="clear:both;width:95%;" class="NavFrame">
<div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Proof of the reverse Hölder inequality</div>
<div class="NavContent" style="text-align:left;display:none;">

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''&thinsp;{{abs|''g''}}<sup>−''q''/''p''</sup>}} almost everywhere. Solving for the absolute value of {{mvar|f}}&thinsp; gives the claim.

</div>
</div>

== Conditional Hölder inequality ==
Let {{math|(Ω,&thinsp;{{mathcal|F}},&thinsp;ℙ)}} be a probability space, {{math|{{mathcal|G}} ⊂ {{mathcal|F}}}} a {{nowrap|sub-[[σ-algebra]]}}, and {{math|''p'', ''q'' ∈}} {{open-open|1,&thinsp;∞}} 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;∞.

<div style="clear:both;width:95%;" class="NavFrame">
<div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Proof of the conditional Hölder inequality</div>
<div class="NavContent" style="text-align:left;display:none;">

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''&thinsp;}} {{=}} 0}} a.s. on the set {{math|{{mset|''U'' {{=}} 0}}}}. Similarly, {{math|{{abs|''Y''&thinsp;}} {{=}} 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>
</div>
</div>

{{more footnotes|date=April 2012}}

==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.

== See also ==
* [[Cauchy–Schwarz inequality]]
* [[Minkowski inequality]]
* [[Jensen's inequality]]
* [[Young's inequality for products]]
* [[Clarkson's inequalities]]
* [[Brascamp–Lieb inequality]]

==Citations==
{{reflist}}

==References==
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*{{citation
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|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
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*{{springer
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*{{Citation
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==External links==
*{{Citation
 | last=Kuttler
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*{{Citation
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|url=http://www.mediafire.com/?1mw1tkgozzu
|year=1982
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*{{Citation
 | last=Tisdell
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 | title=Holder's Inequality
 | publisher=Online video on Dr Chris Tisdell's YouTube channel
 | url=https://www.youtube.com/watch?v=kxQiKaIuyOg
 | year=2012
}}.

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