Zero sets of holomorphic functions in the unit ball: non-radial growth characteristics

Let $f$ be a nonzero holomorphic function in the unit ball $\mathbb{B}$ of the $n$-dimensional complex Euclidean space ${\mathbb{C}}^n$ such that the function $f$ vanishes on the set $\mathsf{Z}\subset \mathbb{B}$ and satisfies the constraint $\mathrm{\mid }f\mathrm{\mid }\le \exp M$ on $\mathbb{B}$, where $M\equiv ̸\pm \mathrm{\infty }$ is $\delta$-subharmonic function on $\mathbb{B}$ with Riesz charge ${\mu }_M$. We give a scale of integral uniform constraints from above on the distribution of the set $\mathsf{Z}$ via the charge ${\nu }_M$ in terms of $(2n-2)$-Hausdorff measure of the set $\mathsf{Z}$, as well as test convex radial functions and $\rho$-subspherical functions on the unit sphere $\mathbb S \subset \mathbb C^n$,whichat$n=1$canbeinterpretedas$2\pi$-periodic$\rho$-trigonometricallyconvexfunctionsontherealaxis$\mathbb R \subset \mathbb C$.