Abstract
The aim of this paper is to introduce new hyperbolic classes of functions, which will be called B-alpha,B- log* and F-log* (p, q, s) classes. Furthermore, we introduce D- metrics space in the hyperbolic type classes B-alpha,B- log* and F-log* (p, q, s). These classes are shown to be complete metric spaces with respect to the corresponding metrics. Moreover, necessary and sufficient conditions are given for the composition operator C-phi to be bounded and compact from B-alpha,B-log* to F-log*(p, q, s) spaces.