A GENERALIZATION OF A RESULT OF CHOA ON ANALYTIC FUNCTIONS WITH HADAMARD GAPS

Title & Authors
A GENERALIZATION OF A RESULT OF CHOA ON ANALYTIC FUNCTIONS WITH HADAMARD GAPS
Stevic Stevo;

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 $\small{f(z)\;=\;{\sum}^{\infty}_{k=1}\;P_{nk}(z)}$ (the homogeneous polynomial expansion of f) satisfying $n_{k+1}/n_{k}{\ge}{\lambda}>1$ for all $\small{k\;{\in}\;N}$, to belong to the weighted Bergman space $$A^p_{\alpha}(B)\;=\;\{f{\mid}{\int}_{B}{\mid}f(z){\mid}^{p}(1-{\mid}z{\mid}^2)^{\alpha}dV(z) < {\infty},\;f{\in}H(B)\}$$. We find a growth estimate for the integral mean $\small{$${\int}_{{\partial}B}{\mid}f(r{\zeta}){\mid}^pd{\sigma}({\zeta})$$^{1/p}}$, and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $\small{H_{p,q,{\alpha}}$(B) and weighted Bergman space on polydisc $\small{A^p_{^{\to}_{\alpha}}(U^n)}$ are also given.
Keywords
Language
English
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