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 ∞ a k z n k where n k + 1 / n k ⩾ λ > 1 for all k ∈ N , to belong to the Bloch-type space B μ ( D ) = f | sup z ∈ 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.