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For a domain Ω in the complex plane C, let H(Ω) denote the space of functions holomorphic in Ω, and let P be a family of functions subharmonic in Ω. Denote by HP(Ω) the class of f ∈ H(Ω) satisfying |f(z)| ≤ Cf exp pf (z), z ∈ Ω, where pf ∈ P and Cf is a constant. The paper is aimed at conditions for a set Λ ⊂ Ω to be included in the zero set of some nonzero function in HP(Ω). In the first part, certain preparatory theorems are established concerning “quenching” the growth of a subharmonic function by adding to it a function of the form log |h|, where h is a nonzero function in H(Ω). The method is based on the balayage of measures and subharmonic functions. Introduction Nevanlinna’s classical ultimate description of the zero sets for the algebra H∞ of bounded holomorphic functions in the unit disk D = {z ∈ C : |z| < 1} (C is the complex plane) and similar results for the classes of Nevanlinna, Nevanlinna–Dzhrbashyan, and Hardy gave rise to a wide range of related studies for a variety of weighted algebras and spaces of holomorphic functions in D. For the topic so broad and rich in results, we have made no attempt to achieve even a minimal coverage of the relevant bibliography. We only refer to the surveys [1] by Shvedenko, [2] by Aleksandrov, [3] by Hedenmalm, and [4] by Colwell, as well as to the treatment of spaces and algebras much similar to those studied here: the monograph [5] by Dzhrbashyan and Shamoyan, the final results [6, 7] by Shamoyan (a far-reaching development of Dzhrbashyan’s work), the results of Horowitz [8] (on algebras of functions of moderate “exponential” or fast growth), the work of Korenblum [9], of Beller and Horowitz [10, 11], of Seip [12, 13], of Bruna and Massaneda [14], of Lueking [15] (where, in particular, stability conditions for a subsequence of zeros for Bergman-type spaces were treated), the joint monograph [16] by Hedenmalm, Korenblum, and Zhu, and the paper [17] by Blasco, Kukuryka, and Nowak (concerning spaces and algebras of functions of slow “power” growth). The Introductions and reference lists in these papers and books can give the reader an idea about the recent state of the art in the topic in question. In the present study, we concentrate on conditions for a sequence of points in a domain Ω ⊂ C to be a subsequence of zeros of a nontrivial function holomorphic in Ω and belonging to a certain class; these classes will be defined by growth restrictions near 2000 Mathematics Subject Classification. Primary 30C15.

Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. I