Re: Normal families, complex analysis question

On Thu, 22 Dec 2005 10:20:16 -0500, "James" <James545@xxxxxxxxx>
wrote:

>Let H be the family of functions h analytic on D = { | z | < 1 } so that
>
>h(D) is contained in C - [-oo,0].
>
>I am trying to show that H is a normal family.
>
>So the image of h is contained in a domain that can be used as a branch of
>log. The function
>
>f (z) = [sqrt(z) - 1] / [sqrt(z) + 1]
>
>maps C - [-oo,0] analytically isomorphically to D.
>
>So I have that | f o h | <= 1
>
>for all h in H. I want to show that H is locally bounded.
>
>But that is all I can get, and it doesn't help me at all. Any thoughts?

You actually have that | f o h | < 1 in D, and this shows
that if K is a compact subset of D then there exists c < 1
such that | f o h | <= c in K. That shows that h|K takes
values in some compact subset of C - [-oo,0].

>James
>


************************

David C. Ullrich
.

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