A Generalization of a Result of Choa on Analytic Functions With Hadamard Gaps

Abstract

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) = ∞ k=1 Pn k (z) (the homogeneous polynomial expansion of f ) satisfying n k+1 /n k ≥ λ > 1 for all k ∈ N, to belong to the weighted Bergman spaceWe find a growth estimate for the integral mean

On Bloch-Type Functions with Hadamard Gaps

“…Stevo Stević 7 Since f has Hadamard gaps and f (m) (z) = ∞ k=1 a k n k (n k − 1)•••(n k − m + 1)z nk−m , it follows that f (m) has Hadamard gaps too. Applying the just proved result to the function f (m) , we obtain that f (m) Note that the result is true for the case of the weighted Bergman space, that is, when p = q, see [12,Corollary 1]. It is also expected that Theorems 1.3 and 1.4 hold for every p ∈ [1;∞] (for the case n = 1, see [13]).…”

Lacunary Series in Weighted HyperholomorphicBp,q(G) Spaces

“…In the past few decades both Taylor and Fourier series expansions were studied by the help of lacunary series (see [4,5,25,27,31,33,34,36] and others). On the other hand there are some characterizations in higher dimensions using several complex variables and quaternion sense (see [3,11,13,16,20,23,29,30]). …”

Theory of Bergman Spaces in the Unit Ball of C^n

“…For example, lacunary series in the Bloch space of the unit disk are described in Anderson-Clunie-Pommerenke [3], lacunary series in weighted Bergman spaces A p α of the unit ball, where α > −1, are described in Stević [53], and lacunary series in Bloch and certain Lipschitz spaces of the unit ball are characterized in Wulan-Zhu [59].…”