Unbounded H\"{o}lder domains

Motivated by the classical bounded H\"{o}lder domains, we introduce the notion of an unbounded simply connected H\"{o}lder domain. We prove analytic and geometric characterizations of those domains with the aid of the spherical metric and the hyperbolic metric. We also study the relation of our definition to the definition of classical bounded H\"{o}lder domains. It turns out that unbounded H\"older domains form a natural class of domains for the study of the Hardy number (which determines the Hardy spaces to which the corresponding Riemann mapping belongs to). As an application of our characterizations, we prove a sharp bound for the Hardy number of an unbounded H\"{o}lder domain.