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

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