# Consequence of Cauchy's Integral Formula

1. Mar 13, 2009

### Nebula

1. The problem statement, all variables and given/known data
Suppose f is analytic inside and on the unit circle |z|=1. Prove that if |f(z)|$$\leq$$M for |z|=1, then |f(0)|$$\leq$$M and |f'(0)|$$\leq$$M.

Finally, what estimate can you give for |f^n(0)|?

2. Relevant equations
This problem is in a section dealing with Cauchy's Integral Formula and it's generalized form.
http://en.wikipedia.org/wiki/Cauchy_integral_formula

3. The attempt at a solution
Im not quite sure how to get started here. Obviously Cauchy's formulas apply to our function f considering its analiticity. I guess I don't really understand the if part of our statement. IE if |f(z)|$$\leq$$M for |z|=1. The magnitude of the function f is bounded??? I don't know what it means "for |z|=1." I'm missing something here. A nudge in the right direction would be appreciated. Thanks!

2. Mar 13, 2009

### Dick

The absolute value of the integral of a function is less or equal to the integral of the absolute value of the function times the length of the curve you are integrating over. The 'a' the Cauchy formula is zero. |z|=1. The function f is bounded on |z|=1 by M. That's a nudge.