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Zeros of holomorphic functions in the unit disk and \(\rho \)-trigonometrically convex functions. (English) Zbl 1430.30005

Summary: 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 \(Z\). We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using \(\rho \)-trigonometrically convex functions.

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

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