INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS AS GENERIC LOG-CONVEX WEIGHTS

Abstract

Let ${\mathcal{H}}ol(B_{d})$ denote the space of holomorphic functions on the unit ball $B_{d}$ of $\mathbb{C}^{d}$, $d\geq 1$. Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\in {\mathcal{H}}ol(B_{d})$ such that the standard integral means $M_{p}(f,r)$ and $w(r)$ are equivalent for any $p$ with $0<p\leq \infty$. We also obtain similar results related to volume integral means.