# Cauchy Schwarz inequality

The Cauchy inequality for finite sums of real numbers is the inequality $$\left(\sum_{k=1}^n a_k b_k\right)^2\leq \sum_{k=1}^n a_k^2 \sum_{k=1}^n b_k^2 .$$ 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 $f(z)$, $$f(z)=\sum_{k=0}^\infty c_k (z-a)^k .$$ 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 .

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