Suppose f is analytic inside and on the unit circle |z|=1. Prove that if |f(z)|[tex]\leq[/tex]M for |z|=1, then |f(0)|[tex]\leq[/tex]M and |f'(0)|[tex]\leq[/tex]M.
Finally, what estimate can you give for |f^n(0)|?
This problem is in a section dealing with Cauchy's Integral Formula and it's generalized form.
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)|[tex]\leq[/tex]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!