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The Hölder inequality for sums. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475142.png" /> be certain sets of complex numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475143.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475144.png" /> is a finite or an infinite set of indices. The following inequality of Hölder is valid:
+
{{TEX|done}}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475145.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475146.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475147.png" />; this inequality becomes an equality if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475148.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h0475149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751410.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751411.png" />. In the limit case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751412.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751413.png" />, Hölder's inequality has the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751414.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751415.png" />, Hölder's inequality is reversed. The converse proposition of Hölder's inequality for sums is also true (M. Riesz): If
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751416.png" /></td> </tr></table>
 
 
 
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751417.png" /> such that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751418.png" /></td> </tr></table>
 
  
 +
The Hölder inequality for sums. Let $\{a_s\}$ and $\{b_s\}$ be certain sets of complex numbers, $s\in S$, where $S$ is a finite or an infinite set of indices. The following inequality of Hölder is valid:
 +
\begin{equation}\label{eq:1}
 +
\Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq \Bigl(\sum\limits_{s\in S}|a_s|^p\Bigr)^{\frac1p}\Bigl(\sum\limits_{s\in S}|b_s|^q\Bigr)^{\frac1q},
 +
\end{equation}
 +
where $p>1$ and $\frac1p + \frac1q =1$; this inequality becomes an equality if and only if $|a_s|^p = C|b_s|^q$, and $\arg(a_sb_s)$ and $C$ are independent of $s\in S$. In the limit case, when $p=1$, $q=+\infty$, Hölder's inequality has the form
 +
\begin{equation*}
 +
\Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq \Bigl(\sum\limits_{s\in S}|a_s|\Bigr)\sup\limits_{s\in S}|b_s|.
 +
\end{equation*}
 +
If $0<p<1$, Hölder's inequality is reversed. The converse proposition of Hölder's inequality for sums is also true (M. Riesz): If
 +
\begin{equation*}
 +
\Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq AB
 +
\end{equation*}
 +
for all $\{a_s\}$ such that
 +
\begin{equation*}
 +
\sum\limits_{s\in S}|a_s|^p \leq A^p,
 +
\end{equation*}
 
then
 
then
 
+
\begin{equation*}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751419.png" /></td> </tr></table>
+
\sum\limits_{s\in S}|b_s|^q \leq B^q.
 +
\end{equation*}
  
 
For sums of a more general form, Hölder's inequality takes the form
 
For sums of a more general form, Hölder's inequality takes the form
 +
\begin{equation}\label{eq:2}
 +
\Bigl|\sum\limits_{s\in S}\rho_sa_{1s}\cdot\dots \cdot a_{ms}\Bigr|\leq \prod\limits_{k=1}^m\Bigl(\sum\limits_{s\in S}\rho_s|a_{ks}|^{p_k}\Bigr)^{\frac{1}{p_k}}\quad \rho_s \geq 0,
 +
\end{equation}
 +
if $\frac{1}{p_1} +\dots + \frac{1}{p_m}=1,\quad p_k>1$, and $1\leq k\leq m$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751420.png" /></td> </tr></table>
+
The Hölder inequality for integrals. Let $S$ be a Lebesgue-measurable set in an $n$-dimensional Euclidean space $\mathbb R^n$ and let the functions
 +
\begin{equation*}
 +
a_k(s) = a_k(s^1,\dots,s^n),\quad 1\leq k\leq m,
 +
\end{equation*}
 +
belong to $L_{p_k}(S)$. The following inequality of Hölder is then valid:
 +
\begin{equation*}
 +
\Bigl|\int\limits_{S}a_1(s)\cdot\dots \cdot a_m(s)\, ds\Bigr|\leq \prod\limits_{k=1}^m\Bigl(\int\limits_{S}|a_k(s)|^{p_k}\Bigr)^{\frac{1}{p_k}}.
 +
\end{equation*}
  
if
+
If $m=p=q=2$, one obtains the [[Bunyakovskii inequality|Bunyakovskii inequality]]. Analogous remarks (concerning the sign and the limit case) as were made for the Hölder inequality
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751421.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
 
 
 
The Hölder inequality for integrals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751422.png" /> be a Lebesgue-measurable set in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751423.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751424.png" /> and let the functions
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751425.png" /></td> </tr></table>
 
 
 
belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751426.png" />, condition
 
 
 
being satisfied. The following inequality of Hölder is then valid:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751427.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751428.png" />, one obtains the [[Bunyakovskii inequality|Bunyakovskii inequality]]. Analogous remarks (concerning the sign and the limit case) as were made for the Hölder inequality
 
  
 
are also valid for the Hölder inequality for integrals.
 
are also valid for the Hölder inequality for integrals.
  
In the Hölder inequality the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751429.png" /> may be any set with an additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751430.png" /> (e.g. a measure) specified on some algebra of its subsets, while the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751431.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751432.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751433.png" />-measurable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751434.png" />-integrable to degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751435.png" />.
+
In the Hölder inequality the set $S$ may be any set with an [[Additive_function | additive function]] $\mu$ (e.g. a measure) specified on some algebra of its subsets, while the functions $a_k(s)$, $1\leq k\leq m$, are $\mu$-measurable and $\mu$-integrable to degree $p_k$.
 
 
The generalized Hölder inequality. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751436.png" /> be an arbitrary set, let a (finite or infinite) functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751437.png" /> be defined on the totality of all positive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751438.png" /> and let this functional satisfy the following conditions: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751439.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751440.png" /> for all numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751441.png" />; c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751442.png" />, then the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751443.png" /> is valid; and d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751444.png" />. If the conditions
 
 
 
are also met, the generalized Hölder inequality is valid for the functional:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047514/h04751445.png" /></td> </tr></table>
+
The generalized Hölder inequality. Let $S$ be an arbitrary set, let a (finite or infinite) functional $\phi:a\to\phi(a)$ be defined on the totality of all positive functions $a:S\to\mathbb R$ and let this functional satisfy the following conditions: a) $\phi(0)=0$; b) $\phi(\lambda a)=\lambda\phi(a)$ for all numbers $\lambda>0$; c) if $0<a(s)\leq b(s)$, then the inequality $\phi(a)\leq \phi(b)$ is valid; and d) $\phi(a+b) \leq \phi(a) + \phi(b)$. If the conditions are also met, the generalized Hölder inequality is valid for the functional:
 +
\begin{equation*}
 +
\phi(|a_1\cdot\dots \cdot a_m|)\leq \prod\limits_{k=1}^m[\phi(|a_k|^{p_k})]^{\frac{1}{p_k}}.
 +
\end{equation*}
  
 
====References====
 
====References====
Line 55: Line 54:
  
 
====Comments====
 
====Comments====
The Bunyakovskii inequality is better known as the Cauchy–Schwarz inequality in the English-language literature.
+
The Bunyakovskii inequality is better known as the [[Cauchy_Schwarz_inequality | Cauchy–Schwarz inequality]] in the English-language literature.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>

Latest revision as of 16:13, 29 November 2012


The Hölder inequality for sums. Let $\{a_s\}$ and $\{b_s\}$ be certain sets of complex numbers, $s\in S$, where $S$ is a finite or an infinite set of indices. The following inequality of Hölder is valid: \begin{equation}\label{eq:1} \Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq \Bigl(\sum\limits_{s\in S}|a_s|^p\Bigr)^{\frac1p}\Bigl(\sum\limits_{s\in S}|b_s|^q\Bigr)^{\frac1q}, \end{equation} where $p>1$ and $\frac1p + \frac1q =1$; this inequality becomes an equality if and only if $|a_s|^p = C|b_s|^q$, and $\arg(a_sb_s)$ and $C$ are independent of $s\in S$. In the limit case, when $p=1$, $q=+\infty$, Hölder's inequality has the form \begin{equation*} \Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq \Bigl(\sum\limits_{s\in S}|a_s|\Bigr)\sup\limits_{s\in S}|b_s|. \end{equation*} If $0<p<1$, Hölder's inequality is reversed. The converse proposition of Hölder's inequality for sums is also true (M. Riesz): If \begin{equation*} \Bigl|\sum\limits_{s\in S}a_sb_s\Bigr| \leq AB \end{equation*} for all $\{a_s\}$ such that \begin{equation*} \sum\limits_{s\in S}|a_s|^p \leq A^p, \end{equation*} then \begin{equation*} \sum\limits_{s\in S}|b_s|^q \leq B^q. \end{equation*}

For sums of a more general form, Hölder's inequality takes the form \begin{equation}\label{eq:2} \Bigl|\sum\limits_{s\in S}\rho_sa_{1s}\cdot\dots \cdot a_{ms}\Bigr|\leq \prod\limits_{k=1}^m\Bigl(\sum\limits_{s\in S}\rho_s|a_{ks}|^{p_k}\Bigr)^{\frac{1}{p_k}}\quad \rho_s \geq 0, \end{equation} if $\frac{1}{p_1} +\dots + \frac{1}{p_m}=1,\quad p_k>1$, and $1\leq k\leq m$.

The Hölder inequality for integrals. Let $S$ be a Lebesgue-measurable set in an $n$-dimensional Euclidean space $\mathbb R^n$ and let the functions \begin{equation*} a_k(s) = a_k(s^1,\dots,s^n),\quad 1\leq k\leq m, \end{equation*} belong to $L_{p_k}(S)$. The following inequality of Hölder is then valid: \begin{equation*} \Bigl|\int\limits_{S}a_1(s)\cdot\dots \cdot a_m(s)\, ds\Bigr|\leq \prod\limits_{k=1}^m\Bigl(\int\limits_{S}|a_k(s)|^{p_k}\Bigr)^{\frac{1}{p_k}}. \end{equation*}

If $m=p=q=2$, one obtains the Bunyakovskii inequality. Analogous remarks (concerning the sign and the limit case) as were made for the Hölder inequality

are also valid for the Hölder inequality for integrals.

In the Hölder inequality the set $S$ may be any set with an additive function $\mu$ (e.g. a measure) specified on some algebra of its subsets, while the functions $a_k(s)$, $1\leq k\leq m$, are $\mu$-measurable and $\mu$-integrable to degree $p_k$.

The generalized Hölder inequality. Let $S$ be an arbitrary set, let a (finite or infinite) functional $\phi:a\to\phi(a)$ be defined on the totality of all positive functions $a:S\to\mathbb R$ and let this functional satisfy the following conditions: a) $\phi(0)=0$; b) $\phi(\lambda a)=\lambda\phi(a)$ for all numbers $\lambda>0$; c) if $0<a(s)\leq b(s)$, then the inequality $\phi(a)\leq \phi(b)$ is valid; and d) $\phi(a+b) \leq \phi(a) + \phi(b)$. If the conditions are also met, the generalized Hölder inequality is valid for the functional: \begin{equation*} \phi(|a_1\cdot\dots \cdot a_m|)\leq \prod\limits_{k=1}^m[\phi(|a_k|^{p_k})]^{\frac{1}{p_k}}. \end{equation*}

References

[1] O. Hölder, "Ueber einen Mittelwerthsatz" Nachr. Ges. Wiss. Göttingen (1889) pp. 38–47
[2] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)
[3] E.F. Beckenbach, R. Bellman, "Inequalities" , Springer (1961)


Comments

The Bunyakovskii inequality is better known as the Cauchy–Schwarz inequality in the English-language literature.

References

[a1] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
How to Cite This Entry:
Hölder inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%C3%B6lder_inequality&oldid=22585
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article