Integral means of holomorphic functions as generic log-convex weights

Let $\mathcal{H}ol(B_d)$ denote the space of holomorphic functions on the unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. 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 $0<p\le\infty$. Also, we obtain similar results related to volume integral means.