On Some Properties and Examples of Nevanlinna Domains

The properties of Nevanlinna domains are considered. These domains arise in the problems of approximation by polyanalytic functions. Several analytic and geometric properties (both new and earlier known) of Nevanlinna domains are described. In particular, a new method for constructing Nevanlinna domains with boundaries belonging to the class $\mathrm C^1$ is proposed, and new examples of such domains whose boundaries do not belong to the class $\mathrm C^{1,\alpha }$ for $\alpha \in (0,1)$ are presented. This method is based on the property of pseudocontinuation of a conformal mapping from the unit disk onto a Nevanlinna domain.