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 $\mathsf{Z}\subset \mathbb{D}$, and satisfies $|f|\le \exp M$ 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.

11 pages