Natural metrics and composition operators in generalized hyperbolic function spaces

In this paper, we define some generalized hyperbolic function classes. We also introduce natural metrics in the generalized hyperbolic (p, alpha)-Bloch and in the generalized hyperbolic Q*(p, s) classes. These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, boundedness and compactness the composition operators C-phi acting from the generalized hyperbolic (p, alpha)-Bloch class to the class Q*(p, s) are characterized by conditions depending on an analytic self-map phi : D --> D.