Prove: Cauchy’s inequality. If f(z) is analytic inside and on a circle C of radius r and center at a, then

where M is a constant such that |f(z)| < M on C, i.e. M is an upper bound of |f(z)| on C.

Proof. Using Cauchy’s integral formula we have

Now

We now utilize that property of line integrals which states that

where |f(z)| M ( i.e. M is an upper bound of |f(z)| on C) and L is the length of C, plus the facts that |z- a| = r and the length of C is 2πr, to obtain