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Difference between revisions of "Cauchy Schwarz inequality"

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m (Camillo.delellis moved page Cauchy inequality to Cauchy Schwarz inequality: More common name)
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The Cauchy inequality for finite sums of real numbers is the inequality
 
The Cauchy inequality for finite sums of real numbers is the inequality
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\begin{equation}
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\left(\sum_{k=1}^n a_k b_k\right)^2\leq \sum_{k=1}^n a_k^2 \sum_{k=1}^n b_k^2 .
 +
\end{equation}
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Proved by A.L. Cauchy (1821); the analogue for integrals is known as the [[Bunyakovskii 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/c/c020/c020880/c0208801.png" /></td> </tr></table>
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The Cauchy inequality is also the name used for an inequality for the [[modulus]] $|f^{(k)}(a)|$ of a derivative of a regular [[analytic function]] $f(z)$ at a [[fixed point]] $a$ of the complex plane $C$, or for the modulus $|c_k|$ of the coefficients of the [[power series]] expansion of $f(z)$,
 
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\begin{equation}
Proved by A.L. Cauchy (1821); the analogue for integrals is known as the [[Bunyakovskii inequality|Bunyakovskii inequality]].
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f(z)=\sum_{k=0}^\infty c_k (z-a)^k .
 
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\end{equation}
The Cauchy inequality is also the name used for an inequality for the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208802.png" /> of a derivative of a regular analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208803.png" /> at a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208804.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208805.png" />, or for the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208806.png" /> of the coefficients of the power series expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c0208807.png" />,
 
 
 
<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/c/c020/c020880/c0208808.png" /></td> </tr></table>
 
 
 
 
These inequalities are
 
These inequalities are
  
 
<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/c/c020/c020880/c0208809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
<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/c/c020/c020880/c0208809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088010.png" /> is the radius of any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088011.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088012.png" /> is regular, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088013.png" /> is the maximum modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088014.png" /> on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088015.png" />. The inequalities (*) occur in the work of A.L. Cauchy (see e.g. ). They directly imply the Cauchy–Hadamard inequality (see ):
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where $r$ is the radius of any disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088011.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088012.png" /> is regular, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088013.png" /> is the maximum modulus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088014.png" /> on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088015.png" />. The inequalities (*) occur in the work of A.L. Cauchy (see e.g. ). They directly imply the Cauchy–Hadamard inequality (see ):
  
 
<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/c/c020/c020880/c02088016.png" /></td> </tr></table>
 
<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/c/c020/c020880/c02088016.png" /></td> </tr></table>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088033.png" /> are the radii of a polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088034.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088035.png" /> is holomorphic, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088036.png" /> is the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088037.png" /> on the distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088038.png" />.
 
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088033.png" /> are the radii of a polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088034.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088035.png" /> is holomorphic, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088036.png" /> is the maximum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088037.png" /> on the distinguished boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088038.png" />.
  
For references see [[Cauchy–Hadamard theorem|Cauchy–Hadamard theorem]].
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For references see [[Cauchy–Hadamard theorem]].
  
  
  
 
====Comments====
 
====Comments====
In Western literature the name Bunyakovskii inequality is rarely used. Both the inequality for finite sums of real numbers, or its generalization to complex numbers (see [[Bunyakovskii inequality|Bunyakovskii inequality]]), and its analogue for integrals are often called the Schwarz inequality or the Cauchy–Schwarz inequality.
 
  
The distinguished boundary of a polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088039.png" /> as above is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088040.png" />.
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In Western literature the name Bunyakovskii inequality is rarely used. Both the inequality for finite sums of real numbers, or its generalization to complex numbers (see [[Bunyakovskii inequality]]), and its analogue for integrals are often called the Schwarz inequality or the Cauchy–Schwarz inequality.
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 +
The distinguished boundary of a [[polydisc]] $U^n$ as above is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020880/c02088040.png" />.

Revision as of 08:57, 31 May 2016


The Cauchy inequality for finite sums of real numbers is the inequality \begin{equation} \left(\sum_{k=1}^n a_k b_k\right)^2\leq \sum_{k=1}^n a_k^2 \sum_{k=1}^n b_k^2 . \end{equation} Proved by A.L. Cauchy (1821); the analogue for integrals is known as the Bunyakovskii inequality.

The Cauchy inequality is also the name used for an inequality for the modulus $|f^{(k)}(a)|$ of a derivative of a regular analytic function $f(z)$ at a fixed point $a$ of the complex plane $C$, or for the modulus $|c_k|$ of the coefficients of the power series expansion of $f(z)$, \begin{equation} f(z)=\sum_{k=0}^\infty c_k (z-a)^k . \end{equation} These inequalities are

(*)

where $r$ is the radius of any disc on which is regular, and is the maximum modulus of on the circle . The inequalities (*) occur in the work of A.L. Cauchy (see e.g. ). They directly imply the Cauchy–Hadamard inequality (see ):

where is the distance from to the boundary of the domain of holomorphy of . In particular, if is an entire function, then at any point ,

For a holomorphic function of several complex variables , , the Cauchy inequalities are

or

where are the coefficients of the power series expansion of :

are the radii of a polydisc on which is holomorphic, and is the maximum of on the distinguished boundary of .

For references see Cauchy–Hadamard theorem.


Comments

In Western literature the name Bunyakovskii inequality is rarely used. Both the inequality for finite sums of real numbers, or its generalization to complex numbers (see Bunyakovskii inequality), and its analogue for integrals are often called the Schwarz inequality or the Cauchy–Schwarz inequality.

The distinguished boundary of a polydisc $U^n$ as above is the set .

How to Cite This Entry:
Cauchy Schwarz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_Schwarz_inequality&oldid=28863
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article