# Cauchy Schwarz inequality

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

The Cauchy inequality for finite sums of real numbers is the inequality

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 of a derivative of a regular analytic function at a fixed point of the complex plane , or for the modulus of the coefficients of the power series expansion of ,

These inequalities are

 (*)

where 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 .