On Bloch-Type Functions with Hadamard Gaps

Abstract

We give some sufficient and necessary conditions for an analytic function $f$ on the unit ball $B$ with Hadamard gaps, that is, for $f\left(z\right)={\displaystyle {\sum }_{k=1}^{\infty }{P}_{n_k}}\left(z\right)$ (the homogeneous polynomial expansion of $f$) satisfying $n_{k+1}/n_k\ge \lambda >1$ for all $k\in \mathbb{N}$, to belong to the space ${ℬ}_p^{\alpha }\left(B\right)=\{f|{sup}_{0<r<1}{\left(1-r^2\right)}^{\alpha }\backslash |ℛf_r{\mathrm{\backslash |}}_p<\infty ,f\in H\left(B\right)\}$, $p=1,2,\mathrm{\infty }$ as well as to the corresponding little space. A remark on analytic functions with Hadamard gaps on mixed norm space on the unit disk is also given.