Composition operators fromℬαtoQKtype spaces

Suppose that ϕ is an analytic self-map of the unit disk Δ. Necessary and sufficient condition are given for the composition operator Cϕf=fοϕ to be bounded and compact from α-Bloch spaces to QK type spaces which are defined by a nonnegative, nondecreasing function k(r) for 0≤r<∞. Moreover, the compactness of composition operator Cϕ from ℬ0 to QK type spaces are studied, where ℬ0 is the space of analytic functions of f with f′∈H∞ and ‖f‖ℬ0=|f(0)|+‖f′‖∞