Cauchy Schwarz inequality

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 . They directly imply the Cauchy-Hadamard inequality (see ): \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 $T^n=\{z\in\mathbb{C}^n\colon\lvert z_\nu-a_\nu\rvert\leq r_\nu,\; \nu=1,\ldots,n\}$.