Journal of the
Korean Mathematical Society JKMS

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 $f(z)=\sum_{k=1}^\infty P_{n_k}(z)$ (the homogeneous polynomial expansion of $f$) satisfying $n_{k+1}/n_k\geq
\lambda>1$ for all $k\in {\bf N},$ to belong to the weighted Bergman space $${\mathcal A}^p_\alpha(B)=\big\{f\;|\; \int_{B}|f(z)|^p (1-|z|^2)^{\alpha}dV(z)<\infty,\; f\in H(B)\big\}.$$ We find a growth estimate for the integral mean $$\left(\int_{\partial B}|f(r\zeta)|^pd\sigma(\zeta)\right)^{1/p},$$ and an estimate for the point evaluations in this class of functions. Similar results on the mixed norm space $H_{p,q,\alpha}(B)$ and weighted Bergman space on polydisc ${\mathcal A}^p_{\vec{\alpha}}(U^n)$ are also given.

Keywords: analytic functions, Hadamard gap, Bergman space, unit ball

MSC numbers: Primary 32A35