  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 ${\sf Z}\subset \mathbb B$ and satisfies the constraint
$|f|\leq \exp M$ on $\mathbb B$, where $M\not\equiv \pm \infty$ 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 ${\sf Z}$ via the charge $\nu_M$ in terms of $(2n-2)$-Hausdorff measure of
the set $\sf Z$, as well as test convex radial functions and
$\rho$-subspherical functions on the unit sphere $\mathbb S \subset \mathbb
C^n$, which at $n=1$ can be interpreted as $2\pi$-periodic
$\rho$-trigonometrically convex functions on the real axis $\mathbb R \subset
\mathbb C$.
