Abstract
All entire functions which transform a class of holomorphic Zygmund-type spaces into weighted analytic Bloch space using the so-called n -generalized superposition operator are characterized in this paper. Moreover, certain specific properties such as boundedness and compactness of the newly defined class of generalized integral superposition operators are discussed and established by using the concerned entire functions.
Highlights
Let D = fz : jzj < 1g be the known concerned open unit disk in C
When the operator Sφ f ∈ Ξ1 for f ∈ Ξ1, we call that φ is induced by the concerned superposition operator from Ξ into Ξ1
For the ν-Bloch space Bν, we have the following: (i) Every bounded concerned sequence ðhnÞ ∈ Bν is uniformly bounded on concerned compact sets (ii) For any concerned sequence ðhnÞ on Bν with khnkBν ⟶ 0, hn − hnð0Þ ⟶ 0 and the convergence is of uniform type on concerned compact sets
Summary
Assume that Ξ and Ξ1 are two concerned metric spaces of holomorphic-type functions both defined on the complex unit disk D. Let φ be a concerned complex-valued function defined on D. The known superposition operator Sφ on the concerned metric space Ξ is here defined by. It should be noted that the graph of the n-generalized superposition operator Sgφ,n is almost closed, but because the new operator is a nonlinear operator, there is no guarantee to establish its boundedness. Let h ∈ HðDÞ; h is said to be in the weighted little ν-type concerned Bloch space Bν,0 when lim νðjzjÞh′ðzÞ = 0: ð5Þ jzj⟶1−. Bν ð19Þ which is comparatively similar to the concerned constructions of the concerned connected domains as the concerned images for specific classes of functions in different analytictype function spaces can be obtained in [4]
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