Difference between revisions of "Cauchy Schwarz inequality"
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| + | ====The Cauchy inequality for finite sums of real numbers==== | ||
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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Proved by A.L. Cauchy (1821); the analogue for integrals is known as the [[Bunyakovskii inequality]]. | Proved by A.L. Cauchy (1821); the analogue for integrals is known as the [[Bunyakovskii inequality]]. | ||
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| − | + | ====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, and its analogue for integrals are often called the Schwarz inequality or the Cauchy-Schwarz inequality. | |
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| − | + | ====The Cauchy inequality for the modulus of a regular analytic function==== | |
| − | + | 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 $\mathbb{C}$, 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 | ||
| + | \begin{equation}\label{eq:1} | ||
| + | \left\lvert f^{(k)}(a)\right\rvert\leq k!\frac{M(r)}{r^k},\quad\lvert c_k\rvert\leq\frac{M(r)}{r^k}, | ||
| + | \end{equation} | ||
| + | where $r$ is the radius of any disc $U=\{z\in\mathbb{C}\colon\lvert z-a\rvert\leq r\}$ on which $f(z)$ is regular, and $M(r)$ is the maximum modulus of $\lvert f(z)\rvert$ on the circle $\lvert z-a\rvert=r$. The inequalities \eqref{eq:1} occur in the work of A.L. Cauchy <ref name="Cauchy" />. They directly imply the [[Cauchy-Hadamard theorem|Cauchy-Hadamard inequality]] (see <ref name="Hadamard" />): | ||
| + | \begin{equation} | ||
| + | \lim_{k\rightarrow\infty}\sup\left(\frac{\left\lvert f^{(k)}(a)\right\rvert}{k!}\right)^{1/k}\leq\frac{1}{d(a,\partial D)}, | ||
| + | \end{equation} | ||
| + | where $d(a,\partial D)$ is the distance from $a$ to the boundary $\partial D$ of the [[domain of holomorphy]] of $f(z)$. In particular, if $f(z)$ is an [[entire function]], then at any point $a\in\mathbb{C}$, | ||
| + | \begin{equation} | ||
| + | \lim_{k\rightarrow\infty}\sup\left(\frac{\left\lvert f^{(k)}(a)\right\rvert}{k!}\right)^{1/k}=0. | ||
| + | \end{equation} | ||
| + | For a holomorphic function $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n>1$, the Cauchy inequalities are | ||
| + | \begin{equation} | ||
| + | \frac{\partial^{k_1+\cdots+k_n}f(a)}{\partial z_1^{k_1}\cdots\partial z_n^{k_n}}\leq k_1!\cdots k_n!\frac{M(r_1,\ldots,r_n)}{r_1^{k_1}\cdots r_n^{k_n}} | ||
| + | \end{equation} | ||
or | or | ||
| + | \begin{equation} | ||
| + | \lvert c_{k_1,\ldots,k_n}\rvert\leq\frac{M(r_1,\ldots,r_n)}{r_1^{k_1}\cdots r_n^{k_n}},\quad a=(a_1,\ldots,a_n)\in\mathbb{C}^n,\quad k_1,\ldots,k_n=0,1,\ldots, | ||
| + | \end{equation} | ||
| + | where $c_{k_1,\ldots,k_n}$ are the coefficients of the power series expansion of $f(z)$: | ||
| + | \begin{equation} | ||
| + | f(z)=\sum_{k_1,\ldots,k_n}^{\infty}c_{k_1,\ldots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}, | ||
| + | \end{equation} | ||
| + | $r_1,\ldots,r_n$ are the radii of a [[polydisc]] $U^n=\{z\in\mathbb{C}^n\colon\lvert z_j-a_j\rvert\leq r_j,\; j=1,\ldots,n\}$ on which $f(z)$ is holomorphic, and $M(r_1,\ldots,r_n)$ is the maximum of $\lvert f(z)\rvert$ on the distinguished boundary of $U^n$. | ||
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| − | + | ====Comments==== | |
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| − | + | The distinguished boundary of a polydisc $U^n$ as above is the set $T^n=\{z\in\mathbb{C}^n\colon\lvert z_\nu-a_\nu\rvert\leq r_\nu,\; \nu=1,\ldots,n\}$. | |
| − | ==== | + | ====References==== |
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| − | + | <references> | |
| + | <ref name="Cauchy">A.L. Cauchy, "Analyse algébrique" , Gauthier-Villars , Leipzig (1894) (German translation: Springer, 1885)</ref> | ||
| + | <ref name="Hadamard">J. Hadamard, "Essai sur l'etude des fonctions données par leur développement de Taylor" ''J. Math. Pures Appl.'' , '''8''' : 4 (1892) pp. 101–186 (Thesis)</ref> | ||
| + | </references> | ||
Latest revision as of 11:56, 10 June 2016
The Cauchy inequality for finite sums of real numbers
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.
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, and its analogue for integrals are often called the Schwarz inequality or the Cauchy-Schwarz inequality.
The Cauchy inequality for the modulus of a regular analytic function
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 $\mathbb{C}$, 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 \begin{equation}\label{eq:1} \left\lvert f^{(k)}(a)\right\rvert\leq k!\frac{M(r)}{r^k},\quad\lvert c_k\rvert\leq\frac{M(r)}{r^k}, \end{equation} where $r$ is the radius of any disc $U=\{z\in\mathbb{C}\colon\lvert z-a\rvert\leq r\}$ on which $f(z)$ is regular, and $M(r)$ is the maximum modulus of $\lvert f(z)\rvert$ on the circle $\lvert z-a\rvert=r$. The inequalities \eqref{eq:1} occur in the work of A.L. Cauchy [1]. They directly imply the Cauchy-Hadamard inequality (see [2]): \begin{equation} \lim_{k\rightarrow\infty}\sup\left(\frac{\left\lvert f^{(k)}(a)\right\rvert}{k!}\right)^{1/k}\leq\frac{1}{d(a,\partial D)}, \end{equation} where $d(a,\partial D)$ is the distance from $a$ to the boundary $\partial D$ of the domain of holomorphy of $f(z)$. In particular, if $f(z)$ is an entire function, then at any point $a\in\mathbb{C}$, \begin{equation} \lim_{k\rightarrow\infty}\sup\left(\frac{\left\lvert f^{(k)}(a)\right\rvert}{k!}\right)^{1/k}=0. \end{equation}
For a holomorphic function $f(z)$ of several complex variables $z=(z_1,\ldots,z_n)$, $n>1$, the Cauchy inequalities are \begin{equation} \frac{\partial^{k_1+\cdots+k_n}f(a)}{\partial z_1^{k_1}\cdots\partial z_n^{k_n}}\leq k_1!\cdots k_n!\frac{M(r_1,\ldots,r_n)}{r_1^{k_1}\cdots r_n^{k_n}} \end{equation} or \begin{equation} \lvert c_{k_1,\ldots,k_n}\rvert\leq\frac{M(r_1,\ldots,r_n)}{r_1^{k_1}\cdots r_n^{k_n}},\quad a=(a_1,\ldots,a_n)\in\mathbb{C}^n,\quad k_1,\ldots,k_n=0,1,\ldots, \end{equation} where $c_{k_1,\ldots,k_n}$ are the coefficients of the power series expansion of $f(z)$: \begin{equation} f(z)=\sum_{k_1,\ldots,k_n}^{\infty}c_{k_1,\ldots,k_n}(z_1-a_1)^{k_1}\cdots(z_n-a_n)^{k_n}, \end{equation} $r_1,\ldots,r_n$ are the radii of a polydisc $U^n=\{z\in\mathbb{C}^n\colon\lvert z_j-a_j\rvert\leq r_j,\; j=1,\ldots,n\}$ on which $f(z)$ is holomorphic, and $M(r_1,\ldots,r_n)$ is the maximum of $\lvert f(z)\rvert$ on the distinguished boundary of $U^n$.
Comments
The distinguished boundary of a polydisc $U^n$ as above is the set $T^n=\{z\in\mathbb{C}^n\colon\lvert z_\nu-a_\nu\rvert\leq r_\nu,\; \nu=1,\ldots,n\}$.
References
Cauchy Schwarz inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_Schwarz_inequality&oldid=38950