Zeros of holomorphic functions in the unit disk and ρ-trigonometrically convex functions

2018/11/03 by Khabibullin, Bulat N., Khabibullin, Farkhat B.
#30C15 #31A05 #31A15 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.10390

Abstract

Let M\/ be a subharmonic function with Riesz measure μ_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⊂ \mathbb D, and satisfies |f| ≤ exp M on \mathbb D. Then restrictions on the growth of μ_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 ρ-trigonometrically convex functions.

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