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| 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]].
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| 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}},
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| :<math>\|fg\|_1 \le \|f\|_p \|g\|_q.</math>
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| 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]].
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| Hölder's inequality holds even if {{math|{{!!}}''fg''{{!!}}<sub>1</sub>}} is 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'')}}.
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| For {{math|''p'', ''q'' ∈}} {{open-open|1, ∞}} and {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}} and {{math|''g'' ∈ ''L<sup>q</sup>''(''μ'')}}, Hölder's inequality becomes an equality if and only if {{math|{{!}}''f'' {{!}}<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'' {{!}}<sup>''p''</sup> {{=}} ''β'' {{!}}''g''{{!}}<sup>''q''</sup>}} {{mvar|μ}}-[[almost everywhere]].
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| 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, ∞}}.
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| Hölder's inequality was first found by {{harvtxt|Rogers|1888}}, and discovered independently by {{harvtxt|Hölder|1889}}.
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| ==Remarks==
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| ===Conventions===
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| The brief statement of Hölder's inequality uses some conventions.
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| * In the definition of Hölder conjugates, {{math|1/ ∞}} means zero.
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| * If {{math|''p'', ''q'' ∈}} {{closed-open|1, ∞}}, then {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup>}} and {{math|{{!!}}''g''{{!!}}<sub>''q''</sup>}} stand for the (possibly infinite) expressions
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| ::<math>\biggl(\int_S |f|^p\,\mathrm{d}\mu\biggr)^{1/p}</math> and <math>\biggl(\int_S |g|^q\,\mathrm{d}\mu\biggr)^{1/q}.</math> | |
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| * If {{math|''p'' {{=}} ∞}}, then {{math|{{!!}}''f'' {{!!}}<sub>∞</sup>}} stands for the [[essential supremum]] of {{math|{{!}}''f'' {{!}}}}, similarly for {{math|{{!!}}''g''{{!!}}<sub>∞</sup>}}.
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| * The notation {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup>}} with {{math|1 ≤ ''p'' ≤ ∞}} is a slight abuse, because in general it is only a [[norm (mathematics)|norm]] of {{mvar|f}}  if {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup>}} 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.
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| * On the right-hand side of Hölder's inequality, 0 times ∞ as well as ∞ times 0 means 0. Multiplying {{math|''a'' > 0}} with ∞ gives ∞.
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| ===Estimates for integrable products===
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| As above, let {{mvar|f}}  and {{mvar|g}} denote measurable real- or complex-valued functions defined on {{mvar|S}}. If {{math|{{!!}}''fg''{{!!}}<sub>1</sup>}} is finite, then the pointwise products of {{mvar|f}}  with {{mvar|g}} and its [[complex conjugate]] function, respectively, are {{mvar|μ}}-integrable, the estimates
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| :<math>\biggl|\int_S f\bar g\,\mathrm{d}\mu\biggr|\le\int_S|fg|\,\mathrm{d}\mu =\|fg\|_1</math>
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| 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
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| :<math>|\langle f,g\rangle| \le \|f\|_2 \|g\|_2\,,</math>
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| 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|{{!!}}''f'' {{!!}}<sub>2</sup>}} and {{math|{{!!}}''g''{{!!}}<sub>2</sup>}} are finite to make sure that the inner product of {{mvar|f}}  and {{mvar|g}} is well defined.
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| We may recover the original inequality (for the case {{math|''p'' {{=}} 2}}) by using the functions {{math|{{!}}''f'' {{!}}}} and {{math|{{!}}''g''{{!}}}} in place of {{mvar|f}}  and {{mvar|g}}.
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| ===Generalization for probability measures===
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| 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
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| :<math>\|fg\|_1 \le \|f\|_p \|g\|_q</math>
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| for all measurable real- or complex-valued functions {{mvar|f}}  and {{mvar|g}} on {{mvar|S}},
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| ==Notable special cases==
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| 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}}.
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| ===Counting measure===
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| In the case of {{mvar|n}}-dimensional [[Euclidean space]], when the set {{mvar|S}} is {{math|{{mset|1, …, ''n''}}}} with the [[counting measure]], we have
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| :<math>\sum_{k=1}^n |x_k\,y_k| \le \biggl( \sum_{k=1}^n |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^n |y_k|^q \biggr)^{\!1/q}
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| \text{ for all }(x_1,\ldots,x_n),(y_1,\ldots,y_n)\in\mathbb{R}^n\text{ or }\mathbb{C}^n.</math>
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| If {{math|''S'' {{=}} ℕ}} with the counting measure, then we get Hölder's inequality for [[sequence space]]s:
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| :<math>\sum\limits_{k=1}^{\infty} |x_k\,y_k| \le \biggl( \sum_{k=1}^{\infty} |x_k|^p \biggr)^{\!1/p\;} \biggl( \sum_{k=1}^{\infty} |y_k|^q \biggr)^{\!1/q}
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| \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>
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| ===Lebesgue measure===
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| If {{mvar|S}} is a measurable subset of {{math|ℝ<sup>''n''</sup>}} with the [[Lebesgue measure]], and {{mvar|f}}  and {{mvar|g}} are measurable real- or complex-valued functions on {{mvar|S}}, then Hölder inequality is
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| :<math>\int_S \bigl| f(x)g(x)\bigr| \,\mathrm{d}x \le\biggl(\int_S |f(x)|^p\,\mathrm{d}x\biggr)^{\!1/p\;} \biggl(\int_S |g(x)|^q\,\mathrm{d}x\biggr)^{\!1/q}.</math>
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| ===Probability measure===
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| For the [[probability space]] {{math|(Ω, {{mathcal|F}}, ℙ)}}, let {{math|'''E'''}} denote the [[expected value|expectation operator]]. For real- or complex-valued [[random variable]]s {{mvar|X}} and {{mvar|Y}} on {{math|Ω}}, Hölder's inequality reads
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| :<math> {\mathbb E}\bigl[|XY|\bigr] \le \bigl({\mathbb E}\bigl[|X|^p\bigr]\bigr)^{1/p}\; \bigl( {\mathbb E}\bigl[|Y|^q\bigr]\bigr)^{1/q}.</math>
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| Let {{math|0 < ''r'' < ''s''}} and define {{math|''p'' {{=}} ''s''/''r''}}. Then {{math|''q'' {{=}} ''p''/(''p'' − 1)}} is the Hölder conjugate of {{mvar|''p''}}. Applying Hölder's inequality to the random variables {{math|{{!}}''X'' {{!}}<sup>''r''</sup>}} and {{math|1<sub>Ω</sub>}}, we obtain
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| :<math>{\mathbb E}\bigl[|X|^r\bigr]\le\bigl({\mathbb E}\bigl[|X|^s\bigr]\bigr)^{r/s}.</math>
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| In particular, if the {{mvar|s}}<sup>th</sup> absolute [[moment (mathematics)|moment]] is finite, then the {{mvar|r}}<sup> th</sup> absolute moment is finite, too. (This also follows from [[Jensen's inequality]].)
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| ===Product measure===
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| 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
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| :<math>S=S_1\times S_2,\quad \Sigma=\Sigma_1\otimes\Sigma_2,\quad \mu=\mu_1\otimes\mu_2,</math>
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| 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%27s_theorem#Tonelli.27s_theorem|Tonelli's theorem]] allows us to rewrite Hölder's inequality using iterated integrals: If {{mvar|f}}  and {{mvar|g}} are {{nowrap|{{math|Σ}}-measurable}} real- or complex-valued functions on the Cartesian product {{mvar|S}}, then
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| :<math>\begin{align}
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| \int_{S_1}&\int_{S_2}|f(x,y)\,g(x,y)|\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\\
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| &\le\biggl(\int_{S_1}\int_{S_2}|f(x,y)|^p\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\biggr)^{\!1/p\;}\biggl(\int_{S_1}\int_{S_2}|g(x,y)|^q\,\mu_2(\mathrm{d}y)\,\mu_1(\mathrm{d}x)\biggr)^{\!1/q}.\end{align}</math>
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| This can be generalized to more than two {{nowrap|σ-finite}} measure spaces.
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| ===Vector-valued functions===
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| 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 {{nowrap|{{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
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| :<math>\begin{align}
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| \int_S&\sum_{k=1}^n|f_k(x)\,g_k(x)|\,\mu(\mathrm{d}x)\\
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| &\le\biggl(\int_S\sum_{k=1}^n|f_k(x)|^p\,\mu(\mathrm{d}x)\biggr)^{\!1/p\;}\biggl(\int_S\sum_{k=1}^n|g_k(x)|^q\,\mu(\mathrm{d}x)\biggr)^{\!1/q}.\end{align}</math>
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| 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
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| :<math>\alpha(|f_1(x)|^p,\ldots,|f_n(x)|^p)=\beta(|g_1(x)|^q,\ldots,|g_n(x)|^q)</math> for {{mvar|μ}}-almost all {{mvar|x}} in {{mvar|S}}.
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| This finite-dimensional version generalizes to functions {{mvar|f}}  and {{mvar|g}} taking values in a [[sequence space]].
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| == Proof of Hölder's inequality ==
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| There are several proofs of Hölder's inequality; the main idea in the following is [[Young's inequality]].
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| If {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup> {{=}} 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|{{!!}}''g''{{!!}}<sub>''q''</sup> {{=}} 0}}. Therefore, we may assume {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup> > 0}} and {{math|{{!!}}''g''{{!!}}<sub>''q''</sup> > 0}} in the following.
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| If {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup> {{=}} ∞}} or {{math|{{!!}}''g''{{!!}}<sub>''q''</sup> {{=}} ∞}}, then the right-hand side of Hölder's inequality is infinite. Therefore, we may assume that {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup>}} and {{math|{{!!}}''g''{{!!}}<sub>''q''</sup>}} are in {{open-open|0, ∞}}.
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| If {{math|''p'' {{=}} ∞}} and {{math|''q'' {{=}} 1}}, then {{math|{{!}}''fg''{{!}} ≤ {{!!}}''f'' {{!!}}<sub>∞</sup> {{!}}''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, ∞}}.
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| Dividing {{mvar|f}}  and {{mvar|g}} by {{math|{{!!}}''f'' {{!!}}<sub>''p''</sup>}} and {{math|{{!!}}''g''{{!!}}<sub>''q''</sup>}}, respectively, we can assume that
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| :<math>\|f\|_p = \|g\|_q = 1.</math>
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| We now use Young's inequality, which states that
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| :<math>a b \le \frac{a^p}p + \frac{b^q}q</math>
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| 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
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| :<math>|f(s)g(s)| \le \frac{|f(s)|^p}p + \frac{|g(s)|^q}q,\qquad s\in S.</math>
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| Integrating both sides gives
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| :<math>\|fg\|_1 \le \frac{1}p + \frac{1}q = 1,</math>
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| which proves the claim.
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| Under the assumptions {{math|''p'' ∈}} {{open-open|1, ∞}} and {{math|{{!!}}''f'' {{!!}}<sub>''p''</sub> {{=}} {{!!}}''g''{{!!}}<sub>''q''</sub> {{=}} 1}}, equality holds if and only if {{math|{{!}}''f'' {{!}}<sup>''p''</sup> {{=}} {{!}}''g''{{!}}<sup>''q''</sup>}} almost everywhere. More generally, if {{math|{{!!}}''f'' {{!!}}<sub>''p''</sub>}} and {{math|{{!!}}''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
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| :<math>\alpha=\|g\|_q^q</math> and <math>\beta=\|f\|_p^p,</math>
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| such that
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| :<math>\alpha |f|^p = \beta |g|^q\,</math> ''μ''-almost everywhere (*).
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| The case {{math|{{!!}}''f'' {{!!}}<sub>''p''</sub> {{=}} 0}} corresponds to {{math|''β'' {{=}} 0}} in (*). The case {{math|{{!!}}''g''{{!!}}<sub>''q''</sub> {{=}} 0}} corresponds to {{math|''α'' {{=}} 0}} in (*).
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| == Extremal equality ==
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| === Statement ===
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| Assume that {{math|1 ≤ ''p'' < ∞}} and let {{mvar|q}} denote the Hölder conjugate. Then, for every {{math|''f'' ∈ ''L<sup>p</sup>''(''μ'')}},
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| :<math>\|f\|_p = \max \Bigl\{ \Bigl| \int_S f g \, \mathrm{d}\mu \Bigr| : g\in L^q(\mu),\,\|g\|_q \le 1 \Bigr\},</math>
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| 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
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| :<math>\|f\|_\infty = \sup \Bigl\{ \Bigl| \int_S f g \,\mathrm{d}\mu \Bigr| : g\in L^1(\mu),\,\|g\|_1 \le 1 \Bigr\}.</math>
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| <div style="clear:both;width:95%;" class="NavFrame">
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| <div class="NavHead" style="background-color:#FFFAF0; text-align:left; font-size:larger;">Proof of the extremal equality</div>
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| <div class="NavContent" style="text-align:left;display:none;">
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| By Hölder's inequality, the integrals are well defined and, for {{math|1 ≤ ''p'' ≤ ∞}},
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| :<math>\Bigl|\int_S fg\,\mathrm{d}\mu\Bigl|\le\int_S|fg|\,\mathrm{d}\mu\le\|f\|_p\,,</math>
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| hence the left-hand side is always bounded above by the right-hand side.
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| Conversely, for {{math|1 ≤ ''p'' ≤ ∞}}, observe first that the statement is obvious when {{math|{{!!}}''f'' {{!!}}<sub>''p''</sub> {{=}} 0}}. Therefore, we assume {{math|{{!!}}''f'' {{!!}}<sub>''p''</sub> > 0}} in the following.
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| If {{math|1 ≤ ''p'' < ∞}}, define {{mvar|g}} on {{mvar|S}} by
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| :<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>
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| By checking the cases {{math|''p'' {{=}} 1}} and {{math|1 < ''p'' < ∞}} separately, we see that {{math|{{!!}}''g''{{!!}}<sub>''q''</sub> {{=}} 1}} and
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| :<math>\int_S f g \, \mathrm{d}\mu = \|f\|_p\,.</math>
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| It remains to consider the case {{math|''p'' {{=}} ∞}}. For {{math|''ε'' ∈}} {{open-open|0, 1}} define
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| :<math>A=\bigl\{x\in S:|f(x)|>(1-\varepsilon)\|f\|_\infty\bigr\}.</math>
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| Since {{mvar|f}}  is measurable, {{math|''A'' ∈ Σ}}. By the definition of {{math|{{!!}}''f'' {{!!}}<sub>∞</sub>}} as the [[essential supremum]] of {{mvar|f}}  and the assumption {{math|{{!!}}''f'' {{!!}}<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
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| :<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>
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| Then {{mvar|g}} is well-defined, measurable and {{math|{{!}}''g''(''x''){{!}} ≤ 1/''μ''(''B'')}} for {{math|''x'' ∈ ''B''}}, hence {{math|{{!!}}''g''{{!!}}<sub>1</sub> ≤ 1}}. Furthermore,
| |
| | |
| :<math>\Bigl|\int_S fg\,\mathrm{d}\mu\Bigr| = \int_B\frac{1-\varepsilon}{\mu(B)}\|f\|_\infty\,\mathrm{d}\mu = (1-\varepsilon)\|f\|_\infty.</math>
| |
| </div>
| |
| </div>
| |
| | |
| === Remarks and examples ===
| |
| * The equality for {{math|''p'' {{=}} ∞}} fails whenever there exists a set {{mvar|A}} in the {{nowrap|σ-field}} {{math|Σ}} with {{math|''μ''(''A'') {{=}} ∞}} that has ''no subset'' {{math|''B'' ∈ Σ}} with {{math|0 < ''μ''(''B'') < ∞}} (the simplest example is the {{nowrap|σ-field}} {{math|Σ}} containing just the empty set and {{mvar|S}}, and the measure {{mvar|μ}} with {{math|''μ''(''S'') {{=}} ∞}}. Then the [[indicator function]] {{math|1<sub>''A''</sup>}} satisfies {{math|{{!!}}1<sub>''A''</sub>{{!!}}<sub>∞</sub> {{=}} 1}}, but every {{math|''g'' ∈ ''L''<sup>1</sup>(''μ'')}} has to be {{mvar|μ}}-almost everywhere constant on {{mvar|A}}, because it is {{nowrap|{{math|Σ}}-measurable}}, and this constant has to be zero, because {{mvar|g}} is {{nowrap|{{mvar|μ}}-integrable}}. Therefore, the above supremum for the indicator function {{math|1<sub>''A''</sub>}} is zero and the extremal equality fails.
| |
| * For {{math|''p'' {{=}} ∞}}, the supremum is in general not attained. As an example, let {{mvar|S}} denote the natural numbers (without zero), {{math|Σ}} the power set of {{mvar|S}}, and {{mvar|μ}} the counting measure. Define {{math|''f'' (''n'') {{=}} (''n'' − 1)/''n''}} for every natural number {{mvar|n}}. Then {{math|{{!!}}''f'' {{!!}}<sub>∞</sub> {{=}} 1}}. For {{math|''g'' ∈ ''L''<sup>1</sup>(''μ'')}} with {{math|0 < {{!!}}''g''{{!!}}<sub>1</sub> ≤ 1}}, let {{mvar|m}} denote the smallest natural number with {{math|''g''(''m'') ≠ 0}}. Then
| |
| | |
| ::<math>\Bigl|\int_S fg\,\mathrm{d}\mu\Bigr| \le \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|{{!!}}''f''<sub>1</sub> + ''f''<sub>2</sub>{{!!}}<sub>''p''</sub> ≤ {{!!}}''f''<sub>1</sub>{{!!}}<sub>''p''</sub> + {{!!}}''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 of Hölder's inequality==
| |
| Assume that {{math|''r'' ∈}} {{open-open|0, ∞}} and {{math|''p''<sub>1</sup>, …, ''p<sub>n</sup>'' ∈ }} {{open-closed|0, ∞}} such that
| |
| | |
| :<math>\sum_{k=1}^n \frac1{p_k}=\frac1r.</math>
| |
| | |
| Then, for all measurable real- or complex-valued functions {{math|''f''<sub>1</sub>, …, ''f<sub>n</sup>''}} defined on {{mvar|S}},
| |
| | |
| :<math>\biggl\|\prod_{k=1}^n f_k\biggr\|_r\le \prod_{k=1}^n\|f_k\|_{p_k}.</math>
| |
| | |
| 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'' ∈}} {{open-open|0, 1}}, contrary to the notation, {{math|{{!!}}.{{!!}}<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</sup> ≤ … ≤ ''p<sub>n</sup>''}}.
| |
| | |
| '''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|{{!}}''f<sub>n</sub>''{{!}}}} and using the induction hypothesis, we get
| |
| :<math>\begin{align} \|f_1\cdots f_n\|_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
| |
| | |
| :<math>p:=\frac{p_n}{p_n-r}</math> and <math>q:=\frac{p_n}r</math>
| |
| | |
| are Hölder conjugates in {{open-open|1, ∞}}. Application of Hölder's inequality gives
| |
| | |
| :<math>\bigl\||f_1\cdots f_{n-1}|^r\,|f_n|^r\bigr\|_1
| |
| \le\bigl\||f_1\cdots f_{n-1}|^r\bigr\|_p\,\bigl\||f_n|^r\bigr\|_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</sup>, …, ''p<sub>n</sup>'' ∈}} {{open-closed|0, ∞}} and let {{math|''θ''<sub>1</sup>, …, ''θ<sub>n</sup>'' ∈}} {{open-open|0, 1}} denote weights with {{math|''θ''<sub>1</sup> + … + ''θ<sub>n</sup>'' {{=}} 1}}. Define {{mvar|p}} as the weighted [[harmonic mean]], i.e.,
| |
| | |
| :<math> \frac 1p= \sum_{k=1}^n \frac {\theta_k}{p_k}.</math>
| |
| | |
| Given a measurable real- or complex-valued function {{mvar|f}}  on {{mvar|S}}, define
| |
| | |
| :<math>f_k=|f|^{\theta_k},\quad k\in\{1,\ldots,n\}.</math>
| |
| | |
| Then by the above generalization of Hölder's inequality,
| |
| | |
| :<math>\|f\|_p=\biggl\|\prod_{k=1}^n f_k\biggr\|_p\le \prod_{k=1}^n \|f_k\|_{p_k/\theta_k}=\prod_{k=1}^n \|f\|_{p_k}^{\theta_k}.</math>
| |
| | |
| In particular, taking
| |
| {{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
| |
| | |
| :<math>\| f\|_{p_\theta}\le \|f\|_{p_1}^\theta \cdot \|f\|_{p_0}^{1-\theta},</math>
| |
| for {{math|''θ'' ∈}} {{open-open|0, 1}} and
| |
| | |
| :<math>\frac 1p_\theta= \frac \theta {p_1}+ \frac {1-\theta}{p_0}.</math>
| |
| <!--P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Presse 1991, [http://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'' ∈}} {{open-open|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\ge\|f\|_{1/p}\,\|g\|_{-1/(p-1)}.</math>
| |
| | |
| If {{math|{{!!}}''fg''{{!!}}<sub>1</sub> < ∞}} and {{math|{{!!}}''g''{{!!}}<sub>−1/(''p'' −1)</sub> > 0}}, then the reverse Hölder inequality is an equality if and only if there exists an {{math|''α'' ≥ 0}} such that
| |
| | |
| :<math>|f| = \alpha|g|^{-p/(p-1)}\,</math> {{mvar|μ}}-almost everywhere.
| |
| | |
| '''Note:''' {{math|{{!!}}''f'' {{!!}}<sub>1/''p''</sub>}} and {{math|{{!!}}''g''{{!!}}<sub>−1/(''p'' −1)</sub>}} are not norms, these expressions are just compact notation for
| |
| | |
| :<math>\biggl(\int_S|f|^{1/p}\,\mathrm{d}\mu\biggr)^{\!p}</math> and <math>\biggl(\int_S|g|^{-1/(p-1)}\,\mathrm{d}\mu\biggr)^{-(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} \bigl\||f|^{1/p}\bigr\|_1 &= \bigl\||fg|^{1/p}\,|g|^{-1/p}\bigr\|_1\\
| |
| &\le \bigl\||fg|^{1/p}\bigr\|_p\,\bigl\||g|^{-1/p}\bigr\|_q
| |
| =\|fg\|_1^{1/p}\,\bigl\||g|^{-1/(p-1)}\bigr\|_1^{(p-1)/p}.\end{align}</math>
| |
| | |
| Raising to the power {{mvar|p}}, rewriting and solving for {{math|{{!!}}''fg''{{!!}}<sub>1</sub>}} gives the reverse Hölder inequality.
| |
| | |
| 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|{{!}}''fg''{{!}} {{=}} ''α'' {{!}}''g''{{!}}<sup>−''q''/''p''</sup>}} almost everywhere. Solving for the absolute value of {{mvar|f}}  gives the claim.
| |
| | |
| </div>
| |
| </div>
| |
| | |
| == Conditional Hölder inequality ==
| |
| Let {{math|(Ω, {{mathcal|F}}, ℙ)}} 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 {{math|Ω}},
| |
| | |
| :<math>\mathbb{E}\bigl[|XY|\big|\,\mathcal{G}\bigr]
| |
| \le
| |
| \bigl(\mathbb{E}\bigl[|X|^p\big|\,\mathcal{G}\bigr]\bigr)^{1/p}
| |
| \,\bigl(\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]\bigr)^{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 0. Multiplying {{math|''a'' > 0}} with ∞ gives ∞.
| |
| | |
| <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)^{1/p},\qquad V=\bigl(\mathbb{E}\bigl[|Y|^q\big|\,\mathcal{G}\bigr]\bigr)^{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|{{!}}''X'' {{!}} {{=}} 0}} a.s. on the set {{math|{{mset|''U'' {{=}} 0}}}}. Similarly, {{math|{{!}}''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}}, {{mvar|V}}, {{math|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)^{\!1/p\;}
| |
| \biggl(\mathbb{E}\biggl[\frac{|Y|^q}{V^q}1_G\biggr]\biggr)^{\!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)^{\!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)^{\!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'', ℂ)}} be the space of all complex-valued functions on {{mvar|S}}. Let {{mvar|N}} be an increasing [[seminorm]] on {{math|''F''(''S'', ℂ)}}, meaning that, for all real-valued functions {{mvar|f}}  and {{mvar|g}} in {{math|''F''(''S'', ℂ)}}, if {{math|''f'' (''s'') ≥ ''g''(''s'') ≥ 0}} for all {{math|''s'' ∈ ''S''}}, then {{math|''N''(''f'' ) ≥ ''N''(''g'')}}. The seminorm is also allowed to attain the value ∞. Then, for all {{math|''p'', ''q'' ∈}} {{open-open|1, ∞}} with {{math|1/''p'' + 1/''q'' {{=}} 1}}, which means that they are conjugate Hölder exponents, and for all complex-valued functions {{math|''f'', ''g'' ∈ ''F''(''S'', ℂ)}},<ref>For a proof see {{harv|Trèves|1967|loc=Lemma 20.1, pp. 205–206}}</ref>
| |
| | |
| :<math>N(|fg|) \le \bigl(N(|f|^p)\bigr)^{1/p} \bigl(N(|g|^q)\bigr)^{1/q}.</math>
| |
| | |
| '''Remark:''' If {{math|(''S'', Σ, ''μ'')}} is a [[measure space]] and {{math|''N''(''f'' )}} is the upper Lebesgue integral of the absolute value {{math|{{!}}''f'' {{!}}}}, then the restriction of {{mvar|N}} to all {{nowrap|{{math|Σ}}-measurable}} functions gives the usual version of Hölder's inequality.
| |
| | |
| ==Citations==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{citation
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| |first=G. H.
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| |last=Hardy
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| |author-link = Godfrey Harold Hardy
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| |first2= J. E.
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| |last2=Littlewood
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| |author2-link=John Edensor Littlewood
| |
| |first3= G.
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| |last3= Pólya
| |
| |author3-link= George Pólya
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| |title=Inequalities
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| |publisher= [[Cambridge University Press]]
| |
| |pages=XII+314
| |
| |year=1934
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| |isbn=0-521-35880-9
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| |jfm=60.0169.01
| |
| |mr=
| |
| |zbl=0010.10703
| |
| }}.
| |
| *{{citation
| |
| |first=O.
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| |last= Hölder
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| |author-link = Otto Hölder
| |
| |title= Ueber einen Mittelwertsatz
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| |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 = [[German language|German]]
| |
| |pages = 38–47
| |
| |jfm = 21.0260.07
| |
| }}. Available at [http://www.digizeitschriften.de/index.php?id=64&L=2 Digi Zeitschriften].
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| *{{springer
| |
| | id = H/h047514
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| | first = L. P.
| |
| | last = Kuptsov
| |
| | title = Hölder inequality
| |
| }}.
| |
| *{{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 = http://www.archive.org/details/messengermathem01unkngoog
| |
| | archiveurl = http://www.archive.org/stream/messengermathem01unkngoog#page/n183/mode/1up
| |
| | archivedate = August 21, 2007
| |
| | jfm = 20.0254.02
| |
| }}.
| |
| *{{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
| |
| }}.
| |
| | |
| ==External links==
| |
| *{{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
| |
| | isbn=
| |
| }}.
| |
| *{{Citation
| |
| |last=Lohwater
| |
| |first=Arthur
| |
| |title=Introduction to Inequalities
| |
| |format=PDF
| |
| |url=http://www.mediafire.com/?1mw1tkgozzu
| |
| |year=1982
| |
| |isbn=
| |
| }}.
| |
| | |
| {{DEFAULTSORT:Holder's inequality}}
| |
| [[Category:Inequalities]]
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| [[Category:Probabilistic inequalities]]
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| [[Category:Functional analysis]]
| |
| [[Category:Articles containing proofs]]
| |