D-metric Spaces and Composition Operators Between Hyperbolic Weighted Family of Function Spaces

Abstract

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.

Keywords

• d-metric spaces
• logarithmic hyperbolic classes
• composition operators