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

Abstract

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.