Convergence of Riemann maps For $\epsilon > 0$, let $K_\epsilon = \{e^{i\theta} : \theta \in [0,\pi-\epsilon]\}$ be an almost-semicircle. Let $D_\epsilon = H \setminus K_\epsilon$, where $H$ is the upper half plane. Let $\phi_\epsilon : D_\epsilon \rightarrow H$ be the Riemann map normalized so that $\lim_{z\rightarrow\infty} \phi_\epsilon(z) -z = 0$. Let $\psi_\epsilon = \phi_\epsilon/\phi_\epsilon'(0)$ on the intersection of $A$ of the unit disk with the upper half plane. I am trying to show that $\psi_\epsilon$ converges uniformly on compact subsets of $A$ to a conformal map from $A$ to $H$ as $\epsilon \rightarrow 0$. 
Intuitively, as $\epsilon \rightarrow 0$, $D_\epsilon$ approaches a domain with two connected components, and an appropriately normalize map should approach a biholomorphism on each. But, I do not have much of an idea as to how to turn this intuition into a proof. Does anyone have any suggestions? Even better, are there any general results about the limits of Riemann maps on domains which are "almost disconnected" which can be applied here?
 A: A standard tool is the Carathéodory kernel theorem. It operates with a notion of the limit of domains with a distinguished interior point; the same point that is also used to normalize the Riemann map. 
So, if you normalize $\phi_\epsilon$ so that $\phi_\epsilon(i/2)=i$, they will converge to a conformal map of $H\cap \{z:|z|<1\}$ onto $H$. And if you normalize them so that $\phi_\epsilon(2i)=i$, they will converge to a conformal map of $H\cap \{z:|z|>1\}$ onto $H$.     
Normalization by a boundary value of a derivative is not as reliable: it gives less control on the map. If you have  a reason to use this particular normalization, comment and I'll try to come up with something. 

Additional details:


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*In Wikipedia, the Carathéodory kernel theorem is stated for maps from the unit disk $D$ to a sequence of domains. It applies just as well to maps from halfplane $H$ to a sequence of domains, because they can composed with a fixed map $D\to H$. 

*So, the theorem applies to the inverses of $\phi_\epsilon$, normalized, for example, by $\phi_\epsilon(i)=z_0$. The kernel of the sequence $\phi_\epsilon^{-1}(H)$ depends on the choice of this normalization: if $|z_0|<1$, it is the upper half-disk, if $|z_0|>1$, it is the upper half-plane minus the half-disk.   

*One can infer the convergence of $\phi_\epsilon$ from the convergence of $\phi_\epsilon^{-1}$. Rouche's theorem and Hurwitz's theorems come to mind as  possible tools here. But it may be more practical to read a more detailed discussion of kernel convergence, either in Geometric theory of functions of a complex variable by Goluzin or in Univalent functions by Duren.
