On the zeros of analytic functions in the disk with a given majorant near its boundary

Abstract

Suppose that λ is an arbitrary positive function from C[0, 1) such that λ(r) → ∞ as r → 1 − 0 and satisfying some growth regularity conditions, A(λ) is the set of all holomorphic functions f in the unit disk for which ln|f(z)| ≤ c · λ(|z|), |z| < 1. In this paper, we establish that there exists a function f ∈ A(λ) with root set {z k } k=1 +∞ such that the sequence {|z k |} k=1 +∞ is the uniqueness set for the class A(λ).