Zeros of holomorphic functions in the unit disk and $\rho$-trigonometrically convex functions

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 ${\sf Z}\subset \mathbb D$,andsatisfies$|f| \leq \exp M$on$\mathbb D$. Then restrictions on the growth of ${\mu }_M$ near the boundary of $D$ imply certain restrictions on the distribution of $\mathsf{Z}$. We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $\rho$-trigonometrically convex functions.