On the Hardy number of Koenigs domains

This work studies the Hardy number for the class of hyperbolic planar domains satisfying Abel's inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that for all regular domains in the above class, the Hardy number is bounded by below by a strictly positive constant. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Estimates on the actual values of the Hardy numbers are further obtained in conjunction with geometric attributes of the domains under review. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disk and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.