Abstract
Let M be a subharmonic function in a domain D ⊂ ℂn with Riesz measure νM, and let Z ⊂ D. As was shown in the first of the preceding papers, if there exists a holomorphic function f ≠ 0 in D such that f(Z) = 0 and |f| ⩽ exp M on D, then one has a scale of integral uniform upper bounds for the distribution of the set Z via νM. The present paper shows that for n = 1 this result "almost has a converse." Namely, it follows from such a scale of estimates for the distribution of points of the sequence Z ≔ {zk}k=1,2,... ⊂ D ⊂ ℂ via νM that there exists a nonzero holomorphic function f in D such that f(Z) = 0 and |f| ⩽ exp M↑r on D, where the function M↑r ⩾ M on D is constructed from the averages of M over circles rapidly narrowing when approaching the boundary of D with a possible additive logarithmic term associated with the rate of narrowing of these circles.
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Text Copyright © The Author(s), 2019. Published in Funktsional’nyi Analiz i Ego Prilozheniya, 2019, Vol. 53, No. 2, pp. 42–58.
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Khabibullin, B.N., Khabibullin, F.B. On the Distribution of Zero Sets of Holomorphic Functions: III. Converse Theorems. Funct Anal Its Appl 53, 110–123 (2019). https://doi.org/10.1134/S0016266319020047
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DOI: https://doi.org/10.1134/S0016266319020047