Published online by Cambridge University Press: 07 August 2017
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.