Abstract
In this paper, we define some new weighted hyperbolic type-classes of functions, which will be called B-alpha,B-log* and Q(p,log)* classes. Furthermore, we introduce quasi-metrics in the hyperbolic B-alpha,B-log* and hyperbolic Q(p,log)* type classes. These classes are shown to be complete quasi-metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for the composition operator C-phi be bounded and compact composition operator from B-alpha,B-log* to Q(p,log)* spaces.