Abstract
In this paper, we study Lipschitz continuous, the boundedness and compactness of the composition operator $C_\phi$ acting between the general hyperbolic Bloch type-classes ${\mathcal{B}}^{*}_{p,\log,\alpha}$ and general hyperbolic Besov-type classes ${F_{p,\log}^{*}(p,q,s)}.$ Moreover, these classes are shown to be complete metric spaces with respect to the corresponding metrics.
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