Re: Two questions in Complex Analysis

From: si (quee0849_at_yahoo.co.uk)
Date: 07/21/04 

Date: Wed, 21 Jul 2004 20:30:39 GMT

2. I think the second question can be solved as follows :
Note that g(z):= conjugate (f (conjugate z)) is holomorphic on |z|>1.
Note that f-g =0 on [1,oo). By the identity theorem its zero everywhere(
there's a limit point of zeros). Thus f=g everywhere. So on (-oo,1]
f= conjugate(f)

Mimsy Boro wrote:
> I have to short question in Complex Analysis which are part of an
> excersize in my course. I'm quite stuck, any hint would be helpful.
>
> 1. Let f(z) = \sum_{n=0}^{\infty}a_{n}z^n be an analytic function in
> an open circle of radius R around 0. It is known that | f(z) | <= M.
> Let z_{0} be the closest 0 of f(z). Prove that:
> |z_{0}| >= R|a_{0}|/(M+|a_{0}|)
>
> 2. Let g(z) be an analytic function that is defined for |z|>1.
> It is known that Im ( f(z) ) = 0 for each z in [1,\infty}.
> Prove that Im( f(z) ) = 0 for each z in (-\infty,1].
>
> Thanks In Advance,

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