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 Ufimsk. Mat. Zh., 2016, Volume 8, Issue 1, Pages 113–126 (Mi ufa318)

Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a prescribed density

V. B. Sherstyukov

National Research Nuclear University MEPhI, Kashirskoe highway, 31 115409, Moscow, Russia

Abstract: In the work we find the minimal value that can be taken by the type of an entire function of order $\rho\in(0,1)$ with zeroes of prescribed upper and lower densities and located in an angle of a fixed opening less than $\pi$. The main theorem generalizes the previous result by the author (the zeroes lie on one ray) and by A. Yu. Popov (only the upper density of zeros was taken into consideration). We distinguish and study in detail the case when the an entire function has a measurable sequence of zeroes. We provide applications of the obtained results to the uniqueness theorems for entire functions and to the completeness of exponential systems in the space of analytic in a circle functions with the standard topology of uniform convergence on compact sets.

Keywords: type of an entire function, upper and lower density of zeroes, uniqueness theorem, completeness of exponential system.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-00281-à The work is supported by RFBR (grant no. 13-01-00281-a).

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English version:
Ufa Mathematical Journal, 2016, 8:1, 108–120

Bibliographic databases:

UDC: 517.547.22
MSC: 30D15

Citation: V. B. Sherstyukov, “Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a prescribed density”, Ufimsk. Mat. Zh., 8:1 (2016), 113–126; Ufa Math. J., 8:1 (2016), 108–120

Citation in format AMSBIB
\Bibitem{She16} \by V.~B.~Sherstyukov \paper Minimal value for the type of an entire function of order $\rho\in(0,1)$, whose zeros lie in an angle and have a~prescribed density \jour Ufimsk. Mat. Zh. \yr 2016 \vol 8 \issue 1 \pages 113--126 \mathnet{http://mi.mathnet.ru/ufa318} \elib{http://elibrary.ru/item.asp?id=25631807} \transl \jour Ufa Math. J. \yr 2016 \vol 8 \issue 1 \pages 108--120 \crossref{https://doi.org/10.13108/2016-8-1-108} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000411731100009} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84994339080} 

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This publication is cited in the following articles:
1. G. G. Braichev, V. B. Sherstyukov, “Tochnye otsenki asimptoticheskikh kharakteristik rosta tselykh funktsii s nulyami na zadannykh mnozhestvakh”, Fundament. i prikl. matem., 22:1 (2018), 51–97
2. V. B. Sherstyukov, “Asimptoticheskie svoistva tselykh funktsii s zadannym zakonom raspredeleniya kornei”, Kompleksnyi analiz. Tselye funktsii i ikh primeneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 161, VINITI RAN, M., 2019, 104–129
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