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

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

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Bibliographic databases:

UDC: 517.53+517.574+517.987.1
MSC: 30C15, 31A05, 28A25, 30C85
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} 

• http://mi.mathnet.ru/eng/faa3597
• https://doi.org/10.4213/faa3597
• http://mi.mathnet.ru/eng/faa/v53/i2/p42

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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