Boundary regularity of Nevanlinna domains and univalent functions in model subspaces

Density of certain polynomial modules

“…This condition holds in all explicit examples of d-Nevanlinna domains which we know. But using results from [1] it is possible to construct an example where v has no zeros. The procedure is as follows.…”

“…We remark that the question whether bounded univalent functions belonging to K Θ exist as well as the boundary behavior of such functions was studied in [10,1].…”

Nevanlinna domains with large boundaries

“…Constructing Nevanlinna domains with irregular (for instance nonanalytic, non-smooth, and even more irregular) boundaries is a rather difficult and delicate problem. It was considered in [23,17,2,26,27]. The detailed account of these results will be given in Section 2 below.…”

“…The first example of N -domain with nowhere analytic (but rather smooth) boundary was constructed in [23]. Later on, several constructions of N -domains with boundaries belonging to the class C 1 , but not to the class C 1,α , α ∈ (0, 1), were obtained in [17] and [2]. Furthermore, it was shown in [2] that Nevanlinna domains may have "almost" non-rectifiable boundaries.…”

“…Later on, several constructions of N -domains with boundaries belonging to the class C 1 , but not to the class C 1,α , α ∈ (0, 1), were obtained in [17] and [2]. Furthermore, it was shown in [2] that Nevanlinna domains may have "almost" non-rectifiable boundaries. The first example of an N -domain with non-rectifiable boundary was constructed in [26].…”