How to characterize the images of disk-algebra functions? It is well known that the continuous images  $f:\mathbf D\to \mathbb C$ of the closed unit disk $\mathbf D$ are exactly the non-void
compact, connected,  locally path connected  sets in $\mathbb C$.
In the Oberwolfach report 6 (2008) it was  asked on page 344 for a geometric/topological characterization of those planar compacta which are images of a disk-algebra function. This problem still seems to be unsolved.    Any ideas would be welcome.
 A: Let $E=f(\overline{D})$, where $f$ is analytic in $D$, continuous in $\overline{D}$, and non-constant. Then $E$ has these properties:
a) Interior of $E$ is a bounded region (connected and not empty),
b) $E$ is the closure of its interior,
c) $E$ is locally path connected.
To prove a) let $w_i=f(z_i), i\in\{1,2\}$ be two interior points of $E$, $z_i\in \overline{D}$.
Let $\gamma$ be a curve in $\overline D$ which belongs to $D$ except possibly
its endpoints, and $\gamma$ connects $z_1,z_2$. The image of $\gamma$ connects
$w_1,w_2$and completely belongs to the interior of $E$: all points except the
endpoints, because $f$ is open in $D$ and endpoints by assumption.
To prove b), let $w_0\in E$. Then $w_0=f(z_0), z_0\in\overline{D}$, and there is a sequence $z_j\to z_0,\; z_j\in D, j\geq 1$. The points $w_j=f(z_j)$ are
in the interior of $E$ and $f(w_j)\to w_0$.
c) is the property of all continuous functions as stated in the question.
Now I suppose that a),b,c) are sufficient. Let $\Omega$ be the interior of $E$.
There is a system of cuts $I_j$ such that removing these cuts will make $\Omega$ simply connected.
Then the conformal map $f:D\to\Omega\backslash\cup I_j$ is the required function. The problem is how to show rigorously that such cuts exist, so that the resulting region has locally connected boundary.
Remark. I suppose that the natural method to make cuts is to condsider
the universal cover of the interior, and take the Dirichlet fundamental region of the associated Fuchsian group. The image of the boundary of this region will give a system of cuts.
