Spaces with normal weights and Hadamard gap series

Abstract

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc ${\mathbb{D}}$ with Hadamard gaps, that is, for ${f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}$ where ${n_{k+1}/n_k\geq\lambda >1 }$ for all ${k\in \mathbb{N}}$, to belong to the weighted-type space ${ H_\mu^{\infty}}$, under some condition posed on the weight function μ. We can define the corresponding little weighted-type space ${H_{\mu,0}^{\infty}}$ and give a criterion for functions to belong to it.