Normal families, complex analysis question

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?

James


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