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 Mat. Sb., 2009, Volume 200, Number 9, Pages 95–126 (Mi msb4505)

The distribution of the zeros of holomorphic functions of moderate growth in the unit disc and the representation of meromorphic functions there

E. G. Kudashevaa, B. N. Khabibullinbc*

a Bashkir State Agricultural University
b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences
c Bashkir State University, Faculty of Mathematics

Abstract: Let $\mathbb D$ be the unit disc in the complex plane $\mathbb C$ and $H$ a class of holomorphic functions in $\mathbb D$ distinguished by a restriction on their growth in a neighbourhood of the boundary of the disc which is stated in terms of weight functions of moderate growth. Some results which describe the sequences of zeros for holomorphic functions in classes $H$ of this type are obtained. The weight functions defining $H$ are not necessarily radial; however the results obtained are new even in the case of radial constraints. Conditions for meromorphic functions in $\mathbb D$ ensuring that they can be represented as a ratio of two functions in $H$ sharing no zeros are investigated.
Bibliography: 28 titles.

Keywords: unit disc, holomorphic function, sequence of zeros, weighed space, representation of a meromorphic function, subharmonic functions, Green's function.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm4505

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English version:
Sbornik: Mathematics, 2009, 200:9, 1353–1382

Bibliographic databases:

UDC: 517.547+517.538+517.574
MSC: Primary 30C15; Secondary 30D30, 30H99

Citation: E. G. Kudasheva, B. N. Khabibullin, “The distribution of the zeros of holomorphic functions of moderate growth in the unit disc and the representation of meromorphic functions there”, Mat. Sb., 200:9 (2009), 95–126; Sb. Math., 200:9 (2009), 1353–1382

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb4505
• https://doi.org/10.4213/sm4505
• http://mi.mathnet.ru/eng/msb/v200/i9/p95

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This publication is cited in the following articles:
1. Chyzhykov I.E., “Asymptotic Behaviour of Pth Means of Analytic and Subharmonic Functions in the Unit Disc and Angular Distribution of Zeros”, Isr. J. Math.
2. F. B. Khabibullin, “Sequences of zeroes of holomorphic functions in weight spaces in the unit disk”, Russian Math. (Iz. VUZ), 54:3 (2010), 88–90
3. F. B. Khabibullin, “Ustoichivost (pod)posledovatelnostei nulei dlya klassov golomorfnykh funktsii umerennogo rosta v edinichnom kruge”, Ufimsk. matem. zhurn., 3:3 (2011), 152–163
4. E. A. Sevast'yanov, “On zeros of functions rapidly growing in generalized Bergman spaces”, Russian Math. (Iz. VUZ), 61:11 (2017), 40–52
5. B. N. Khabibullin, A. P. Rozit, “On the Distribution of Zero Sets of Holomorphic Functions”, Funct. Anal. Appl., 52:1 (2018), 21–34
6. B. N. Khabibullin, A. P. Rozit, E. B. Khabibullina, “Poryadkovye versii teoremy Khana—Banakha i ogibayuschie. II. Primeneniya v teorii funktsii”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 93–135
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