Abstract
In this paper, we define some new hyperbolic classes of functions, which will be called B alpha ,log* and Flog*(p,q,s) classes. Then, we introduce D-metrics in these defined hyperbolic classes. Also, these classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for a composition operator C> to be bounded and compact from B alpha ,log* to Flog*(p,q,s).