Hello all,
Let $\mathbb{H}$ be the upper half plane, and $i+\mathbb{H}$ be all points with imaginary part greater than 1. Suppose that $U$ is a domain with a nice boundary and with $i+\mathbb{H} \subseteq U \subseteq \mathbb{H}$. Let $f(z)$ be the conformal map from $\mathbb{H}$ to $U$ which fixes $2i$, which extends continuously to the real axis, and which maps $0$ to a point on the intersection of the imaginary axis and the boundary of $U$. I would like to show that there is a constant $K$ such that
$|f(z)| < K(1+|z|) $
for all $z \in \mathbb{H}$. I think that this is true because it is true if $U = i+\mathbb{H}$(so $f(z) = i+ z/2$) and if $U=\mathbb{H}$(so $f(z)=z$), so I am hoping that there is some kind of "pinch" theorem which works here, but I don't see how to get at this. If it helps, we can also assume that the boundary of $U$ gets arbitrarily close to the real axis near infinity, or even that it is something nice like the graph of $y=1/(1+x^2)$ if that helps. I hope that there is some kind of general theory that someone can point me to. Thanks.
Greg
