Equality holding in Cauchy's Inequality

Hello, this was a problem on an exam:

From Cauchy's Integral Formula, we get Cauchy's Inequality .

Prove that equality holds if and only if for some with .

Proving is easy, just take the modulus of the nth derivative of f.

How do we prove the other direction?

Thanks!

Re: Equality holding in Cauchy's Inequality

Re: Equality holding in Cauchy's Inequality

Ah, From Cauchy's Integral Formula and Inequality.

f is analytic on a ball of radius R

|f(z)| __<__ M for all z in the ball

Re: Equality holding in Cauchy's Inequality

So the statement is:

Let . Let satisfy .

Then , such that .

Correct?

Are you sure it isn't supposed to be ?

Re: Equality holding in Cauchy's Inequality

What I wrote is what was on the exam, but I think that was justified by assuming that z_{0} = 0.

And yes, what you have is correct, as far as I understood the problem