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An equality in mathematical analysis, established by V.Ya. Bunyakovskii [[#References|[1]]] for square-integrable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177702.png" />:
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{{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/b/b017/b017770/b0177703.png" /></td> </tr></table>
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An inequality in mathematical analysis, established by V.Ya. Bunyakovskii <ref name="Bounjakowsky" /> for square-integrable functions $ f $ and $ g $ :
 +
 
 +
\[
 +
\left[ \int_{a}^{b}f(x)g(x)\,dx\right]^2 \le \int_{a}^{b}f^2(x)\,dx \int_{a}^{b}g^2(x)\,dx.
 +
\]
  
 
This inequality is analogous to Cauchy's algebraic inequality
 
This inequality is analogous to Cauchy's algebraic 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/b/b017/b017770/b0177704.png" /></td> </tr></table>
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\[
 +
(a_1 b_1 + \dots + a_n b_n)^2 \le (a_1^2 + \dots + a_n^2)(b_1^2 + \dots + b_n^2).
 +
\]
 +
 
 +
The Bunyakovskii inequality is also known as the Schwarz inequality; however, Bunyakovskii published his study as early as 1859, whereas in H.A. Schwarz' work this inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).
  
<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/b/b017/b017770/b0177705.png" /></td> </tr></table>
 
  
The Bunyakovskii inequality is also known as the Schwarz inequality; however, Bunyakovskii published his study as early as 1859, whereas in H.A. Schwarz' work this inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. [V.Ya. Bunyakovskii] Bounjakowsky,  "Sur quelques inegalités concernant les intégrales aux différences finis"  ''Mem. Acad. Sci. St. Petersbourg (7)'' , '''1'''  (1859)  pp. 9</TD></TR></table>
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<references>
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<ref name="Bounjakowsky">W. [V.Ya. Bunyakovskii] Bounjakowsky,  "Sur quelques inegalités concernant les intégrales aux différences finis"  ''Mem. Acad. Sci. St. Petersbourg (7)'' , '''1'''  (1859)  pp. 9</ref>
 +
</references>
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2. W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)
 +
 
  
  
  
 
====Comments====
 
====Comments====
In Western literature this inequality is often called the Cauchy inequality, or the Cauchy–Schwarz inequality. Its generalization to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177707.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177708.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b0177709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b01777010.png" />, is called the [[Hölder inequality|Hölder inequality]].
 
  
Cauchy's algebraic inequality stated above holds for real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b01777011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b01777012.png" />. For complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b01777013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017770/b01777014.png" />, it reads
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In Western literature this inequality is often called the Cauchy inequality, or the [[Cauchy Schwarz inequality]]. Its generalization to a function $ f $ in $ L_p $ and a function $ g $ in $ L_q $, $ 1/p + 1/q = 1 $, is called 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/b/b017/b017770/b01777015.png" /></td> </tr></table>
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Cauchy's algebraic inequality stated above holds for real numbers $ a_i, b_i, \quad i = 1, \dots, n $. For complex numbers $ a_i, b_i, \quad i = 1, \dots, n$, it reads
  
<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/b/b017/b017770/b01777016.png" /></td> </tr></table>
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\[
 +
\left| a_1 \overline{b_1} + \dots + a_n \overline{b_n}\right|^2 \le (|a_1^2| + \dots + |a_n^2|) \cdot (|b_1^2| + \dots + |b_n^2|).
 +
\]
  
 
It has a generalization analogous to the Hölder inequality.
 
It has a generalization analogous to the Hölder inequality.
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
 

Latest revision as of 09:09, 31 May 2016


An inequality in mathematical analysis, established by V.Ya. Bunyakovskii [1] for square-integrable functions $ f $ and $ g $ :

\[ \left[ \int_{a}^{b}f(x)g(x)\,dx\right]^2 \le \int_{a}^{b}f^2(x)\,dx \int_{a}^{b}g^2(x)\,dx. \]

This inequality is analogous to Cauchy's algebraic inequality

\[ (a_1 b_1 + \dots + a_n b_n)^2 \le (a_1^2 + \dots + a_n^2)(b_1^2 + \dots + b_n^2). \]

The Bunyakovskii inequality is also known as the Schwarz inequality; however, Bunyakovskii published his study as early as 1859, whereas in H.A. Schwarz' work this inequality appeared as late as 1884 (without any reference to the work of Bunyakovskii).


References

  1. W. [V.Ya. Bunyakovskii] Bounjakowsky, "Sur quelques inegalités concernant les intégrales aux différences finis" Mem. Acad. Sci. St. Petersbourg (7) , 1 (1859) pp. 9

2. W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953)



Comments

In Western literature this inequality is often called the Cauchy inequality, or the Cauchy Schwarz inequality. Its generalization to a function $ f $ in $ L_p $ and a function $ g $ in $ L_q $, $ 1/p + 1/q = 1 $, is called the Hölder inequality.

Cauchy's algebraic inequality stated above holds for real numbers $ a_i, b_i, \quad i = 1, \dots, n $. For complex numbers $ a_i, b_i, \quad i = 1, \dots, n$, it reads

\[ \left| a_1 \overline{b_1} + \dots + a_n \overline{b_n}\right|^2 \le (|a_1^2| + \dots + |a_n^2|) \cdot (|b_1^2| + \dots + |b_n^2|). \]

It has a generalization analogous to the Hölder inequality.

How to Cite This Entry:
Bunyakovskii inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bunyakovskii_inequality&oldid=18097
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article