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 $f(z) = \sum_{k=1}^{\infty} P_{n_k} (z)$ (the homogeneous polynomial expansion of $f$) satisfying $n_{k+1}/n_{k} \geq \lambda \gt 1$ for all $k\in \mathbb{N}$, to belong to the space $\mathfrak{B}_p^{\alpha} (B) = \{ f | \sup_{0\lt r \lt 1} (1-r^2)^{\alpha} \| \mathfrak{R} f_r \|_p \lt \infty , f\in H(B) \}$, $p=1,2,\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.