On the Bergman number of hyperbolic domains

Dimitrios Betsakos; Nikolaos Karamanlis

On the Bergman number of hyperbolic domains

Complex Variables 2026-07-15 v1

Authors: Dimitrios Betsakos , Nikolaos Karamanlis

Abstract

We construct a domain $D$ in the plane whose Hardy and Bergman numbers satisfy $0<h(D)<b(D)<+\infty$ . We also calculate the Bergman number for certain classes of domains having countable complement in the plane. Finally, we investigate some of the implications of our analysis in the theory of iteration of holomorphic self-maps of the unit disk. Our methods rely on estimates for the hyperbolic metric and a recent related result that connects the Bergman number with the hyperbolic metric.

Keywords

Bergman number hyperbolic domain Hardy number holomorphic self-map iteration theory hyperbolic metric

Cite

@article{arxiv.2607.14265,
  title  = {On the Bergman number of hyperbolic domains},
  author = {Dimitrios Betsakos and Nikolaos Karamanlis},
  journal= {arXiv preprint arXiv:2607.14265},
  year   = {2026}
}