A generalization of a result of choa on analytic functions with hadamard gaps

Abstract

In this paper we obtain a sufficient and necessary condition for an analytic function f on the unit ball B with Hadamard gaps, that is, for f(z) = Σk=1∞ Pnk (z) (the homogeneous polynomial expansion of f) satisfying nk+1/nk ≥ λ > 1 for all k ε N, to belong to the weighted Bergman space Aαp(B) = {f | ∫B |f(z)|p(1 - |z|2)αdV(z) < ∞, f ε H(B)}. We find a growth estimate for the integral mean (∫∂B|f(rζ) |pdσ(ζ))1/p, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space Hp,q,α(B) and weighted Bergman space on polydisc A α→p(Un) are also given.