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Funktsional. Anal. i Prilozhen., 2019, Volume 53, Issue 2, Pages 42–58 (Mi faa3597)  

This article is cited in 1 scientific paper (total in 1 paper)

On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems

B. N. Khabibullin*, F. B. Khabibullin

Bashkir State University, Ufa

Abstract: Let $M$ be a subharmonic function on a domain $D\subset \mathbb C^n$ with Riesz measure $\nu_M$, ${\mathsf Z} \subset D$. As was shown in the first of the preceding articles, if there exists a holomorphic function $ f\neq 0 $ on $D$, $f ({\mathsf Z}) = 0$, $|f|\leq \exp M$ on $D$, then there is some scale of integral uniform estimates from above of the distribution of the set $\mathsf Z$ in terms of $\nu_M$. In this article we show that for $n = 1$ this result is “almost invertible”. From such scale estimates of the distribution of points of the sequence ${\mathsf Z}:= \{{\mathsf z} _k \}_{k = 1,2, … \subset D \subset \mathbb C$ by $\nu_M$ it follows that there exists a nonzero holomorphic function $f$ in $D$, $f (\mathsf Z) =0$, $|f| \leq \exp M^{\uparrow}$ on $D$, where the function $ M^{\uparrow} \geq M$ on $D$ is constructed by averaging of $M$ in rapidly convergent disks as we approach the boundary of the domain $D$ with some possible additive logarithmic component associated with the rate of narrowing of these disks.

Keywords: holomorphic function, sequence of zeros, subharmonic function, Jensen measure, test function, balayage

Funding Agency Grant Number
Russian Science Foundation 18-11-00002
Russian Foundation for Basic Research 18-51-06002

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/faa3597

Full text: PDF file (376 kB)
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Bibliographic databases:

UDC: 517.53+517.574+517.987.1
MSC: 30C15, 31A05, 28A25, 30C85
Received: 12.07.2018
Revised: 12.07.2018
Accepted:04.02.2019

Citation: B. N. Khabibullin, F. B. Khabibullin, “On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems”, Funktsional. Anal. i Prilozhen., 53:2 (2019), 42–58

Citation in format AMSBIB
\Bibitem{KhaKha19}
\by B.~N.~Khabibullin, F.~B.~Khabibullin
\paper On the Distribution of Zero Sets of Holomorphic Functions. III. Inversion Theorems
\jour Funktsional. Anal. i Prilozhen.
\yr 2019
\vol 53
\issue 2
\pages 42--58
\mathnet{http://mi.mathnet.ru/faa3597}
\crossref{https://doi.org/10.4213/faa3597}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3950327}
\elib{https://elibrary.ru/item.asp?id=37298260}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. B. Menshikova, B. N. Khabibullin, “A criterion for the sequence of roots of holomorphic function with restrictions on its growth”, Russian Math. (Iz. VUZ), 64:5 (2020), 49–55  mathnet  crossref  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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