The aim of this paper is to introduce new hyperbolic classes of functions, which will be called \({\mathcal{B}}^{*} _{\alpha,\;\log}\) and \({ F ^{*}_{\log}}(p,q,s)\) classes. Furthermore, we introduce \(D\)-metrics space in the hyperbolic type classes \({\mathcal{B}}^{*} _{\alpha,\;\log}\) 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 \({\mathcal{B}}^{*}_{\alpha,\;\log}\) to \({F ^{*}_{\log}}(p,q,s)\) spaces.