Bloch-type functions with Hadamard gaps

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