  Let $M$\/ be a subharmonic function with Riesz measure $\mu_M$ on the unit
disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero
holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset
\mathbb D$, and satisfies $|f| \leq \exp M$ on $\mathbb D$. Then restrictions
on the growth of $\mu_M$ near the boundary of $D$ imply certain restrictions on
the distribution of $\sf Z$. We give a quantitative study of this phenomenon in
terms of special non-radial test functions constructed using
$\rho$-trigonometrically convex functions.
