Growth properties and sequences of zeros of analytic functions in spaces of dirichlet type

Abstract

For 0 < p < ∞ we let Dp-1p denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ ℂ : |z| < 1} and satisfy ∫Δ(1 - |z|) p-1 | f1(z)|p dx dy < ∞. The spaces Dp-1p are closely related to Hardy spaces. We have, D p-1p ⊂ Hp, if 0 < p ≤ 2, and H p ⊂ Dp-1p, if 2 ≤ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of D p-1p -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets. © 2006 Australian Mathematical Society.