Bloch-type functions with Hadamard gaps

Abstract

We give a sufficient and a necessary condition for an analytic function f on the unit disc D with Hadamard gaps, that is, for f (z) = ∑k = 1∞ ak znk where nk + 1 / nk ≥ λ > 1 for all k ∈ N, to belong to the Bloch-type space Bμ (D) = fenced(f | supz ∈ D μ (z) | f′ (z) | < ∞, f ∈ H (D)), as well as to the corresponding little Bloch-type space, under some conditions posed on the weight function μ. The main results are applied to a Bloch-type space of interest.