3 Added an example; edited body

Yes. This is a classical result of the geometric function theory due to Carathéodory.

The theorem and a fairly straightforward proof can be found, for instance, in the Hurwitz-Courant Funktionentheorie (in the part written by Courant).

Edit. Theory of Functions of a Complex Variable by Markushevich might be a more accessible reference (see Volume 3, Theorem 2.1).

Edit 2. By the way there exist sequences of domains $\{G_n\subset\mathbb C:\ n\in\mathbb N\}$ such that $$\limsup_{n\to\infty}\ \mbox{dist}(\partial G_n, \partial G)>0$$ but the corresponding Riemann maps $\phi_n:\mathbb D\to G_n$ converge uniformly in $\mathbb D$ to the Riemann map $\phi:\mathbb D\to G$.

For example, let $G_n$ be a union of two disjoint rectangles $G'$ and $G''$ connected with a 'thin' rectangle of the fixed length $l$ and width $h_n=1/n$ (see the picture below).

Let $z_0\in G'$. Then the sequence of the conformal maps $f_n:G_n\to\mathbb D$ which satisfy the conditions $$f_n(z_0)=0,\qquad f'_n(z_0)>0,$$ converges uniformly in $G'$ to the conformal map $f:G'\to\mathbb D$ satisfying the same condition. Moreover, the sequence of the inverse maps $\phi_n:\mathbb D\to G_n$ converges uniformly in $\mathbb D$ to $\phi=f^{-1}$. The general Carathéodory theorem gives a criterion of the convergence of the Riemann maps in terms of the corresponding domains.

Yes. This is a classical result of the geometric function theory due to Carathéodory.

The theorem and a fairly straightforward proof can be found, for instance, in the Hurwitz-Courant Funktionentheorie (in the part written by Courant).

Edit. Theory of Functions of a Complex Variable by Markushevich might be a more accessible reference (see Volume 3, Theorem 2.1).

1

Yes. This is a classical result of the geometric function theory.

The theorem and a fairly straightforward proof can be found, for instance, in the Hurwitz-Courant Funktionentheorie (in the part written by Courant).