The model spaces are subspaces of the Hardy space $H^2$ (in the unit disk $\mathbb D$), invariant under the backward shift operator. Any model space has the form $K_\varTheta=H^2\ominus\varTheta H^2$, where $\varTheta$ is an inner function.

It is planned to discuss three problems related with existence and boundary behavior of univalent functions belonging to model spaces:

To describe inner functions $\varTheta$, such that the model space $K_\varTheta$ contains bounded univalent functions.

Let $f\in K_\varTheta$ be bounded and univalent function in $\mathbb D$, and let $G=f(\mathbb D)$. How large can be the (accessible) boundary of $G$?

Let $R$ be a univalent in $\mathbb D$ rational function of a degree $n$, having their poles outside $\overline{\mathbb D}$. How the quantity

$\displaystyle\int_{\mathbb T}|R'(z)|\,|dz|$ may grow as $n\to\infty$?

The main motivation to study these questions is related to the fact, that the class consisting of all bounded univalent functions $f$ in $\mathbb D$ such that $f\in K_\varTheta$ for some inner function $\varTheta$ is exactly the class of all conformal mappings from $\mathbb D$ onto Nevanlinna domains. Recall that a bounded simply connected domain $G$ in $\mathbb C$ is called a Nevanlinna domain if there exist two functions $u,v\in H^\infty(G)$, such that the equality $\overline{z}=u(z)/v(z)$ holds almost everywhere on $\partial G$ in the sense of conformal mappings. The concept of a Nevanlinna domain have appeared very naturally in problems of approximating functions by polyanalytic polynomials. This concept played a crucial role in recent progress in studies of this approximation problem.