Complex Analysis

[bballife1508]

Suppose f is analytic inside |z|=1. Prove that if |f(z)| is less than or equal to M for |z|=1, then |f(0)| is less than or equal M and |f'(0)| is less than or equal to M.

I'm really stuck here on how to approach this problem. Help PLZ!

 

[tkhunny]

Well, your only clue is that it is "analytic". Please describe what that defintion does for us.

Note: Did your instructor tell you that this is an AWESOME result? If it is analytic and bounded on the boundary, this function is very well behaved inside the boundary. What a wonderful result. Doesn't it give you chills?!