Proving Analytic Function Bounds: Complex Analysis Help and Tips

[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!

 

[snipez90]

If you know the values of an analytic function f on the boundary of the disk, then you can compute f(a) for points a in the interior of the disk via Cauchy's formula. Since you have a bound for the values of f on the boundary of the unit circle, you can estimate the integral representation of f(0) and f'(0) using the generalized Cauchy formula.

If you've already seen Cauchy's estimate (sometimes called Cauchy's inequalities), then apply that directly. But estimating |f(0)| and |f'(0)| via Cauchy's formula is basically rederiving those inequalities in a special case.

 

[bballife1508]

I am not quite sure how to apply Cauchy's estimates to this...

 

[snipez90]

Actually, this is an easier application of Cauchy's estimates than the problem in the other thread with parts a)-c). Look at the actual Cauchy inequality I wrote down in the other thread. If you still don't understand, explain specifically which part of the inequality you don't understand and I'll try to help.