The Bergman and the growth numbers of domains and hyperbolic geometry

In this article we completely characterize the Bergman number of general domains.

Namely, we prove that the Bergman number of a hyperbolic unbounded domain can be calculated in terms of the asymptotic behavior of its hyperbolic metric near infinity.

To obtain the proof of this result we first work in the setting of growth spaces, defining the growth number of a domain.

Later, we prove an equality relating the Bergman number and the growth number of domains.

We also provide examples of domains with prescribed Bergman number which have zero Hardy number, solving a question posed previously by Betsakos and Cruz-Zamorano.

At the end, a similar idea is treated for the case of Bloch-type spaces.