On the Hardy number of Koenigs domains

This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.