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Izv. RAN. Ser. Mat., 2011, Volume 75, Issue 1, Pages 3–28 (Mi izv4104)  

This article is cited in 17 scientific papers (total in 17 papers)

On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros

G. G. Braichev, V. B. Sherstyukov

National Engineering Physics Institute "MEPhI"

Abstract: We find the greatest lower bound for the type of an entire function of order $\rho\in(0,1)$ whose sequence of zeros lies on one ray and has prescribed lower and upper $\rho$-densities. We make a thorough study of the dependence of this extremal quantity on $\rho$ and on properties of the distribution of zeros. The results are applied to an extremal problem on the radii of completeness of systems of exponentials.

Keywords: extremal problems, type of entire function, upper and lower densities of zeros, completeness of systems of exponentials.


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English version:
Izvestiya: Mathematics, 2011, 75:1, 1–27

Bibliographic databases:

UDC: 517.547.22
MSC: 30D20, 30C15, 30D15, 46B15
Received: 06.04.2009
Revised: 31.08.2009

Citation: G. G. Braichev, V. B. Sherstyukov, “On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros”, Izv. RAN. Ser. Mat., 75:1 (2011), 3–28; Izv. Math., 75:1 (2011), 1–27

Citation in format AMSBIB
\by G.~G.~Braichev, V.~B.~Sherstyukov
\paper On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros
\jour Izv. RAN. Ser. Mat.
\yr 2011
\vol 75
\issue 1
\pages 3--28
\jour Izv. Math.
\yr 2011
\vol 75
\issue 1
\pages 1--27

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    This publication is cited in the following articles:
    1. G. G. Braichev, O. V. Sherstjukova, “The Greatest Possible Lower Type of Entire Functions of Order $\rho\in(0;1)$ with Zeros of Fixed $\rho$-Densities”, Math. Notes, 90:2 (2011), 189–203  mathnet  crossref  crossref  mathscinet  isi
    2. G. G. Braichev, V. B. Sherstyukov, “On the Growth of Entire Functions with Discretely Measurable Zeros”, Math. Notes, 91:5 (2012), 630–644  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. G. G. Braichev, “Tochnye otsenki tipov tseloi funktsii poryadka $\rho\in(0;1)$ s nulyami na luche”, Ufimsk. matem. zhurn., 4:1 (2012), 29–37  mathnet
    4. O. V. Sherstyukova, “Ob ekstremalnom tipe tseloi funktsii poryadka menshe edinitsy s nulyami fiksirovannykh plotnostei i shaga”, Ufimsk. matem. zhurn., 4:1 (2012), 161–165  mathnet
    5. G. G. Braichev, “The least type of an entire function of order $\rho\in(0,1)$ having positive zeros with prescribed averaged densities”, Sb. Math., 203:7 (2012), 950–975  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Braichev G.G., “Sharp bounds for the type of an entire function of order less than 1 whose zeros are located on a ray and have given averaged densities”, Dokl. Math., 86:1 (2012), 559–561  crossref  mathscinet  zmath  isi  elib  elib  scopus
    7. A. Yu. Popov, “Development of the Valiron–Levin theorem on the least possible type of entire functions with a given upper $\rho$-density of roots”, Journal of Mathematical Sciences, 211:4 (2015), 579–616  mathnet  crossref
    8. G. G. Braichev, “Sharp Estimates of Types of Entire Functions with Zeros on Rays”, Math. Notes, 97:4 (2015), 510–520  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. O. V. Sherstyukova, “O naimenshem tipe tselykh funktsii poryadka $\rho\in(0,1)$ s nulyami na luche”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:4 (2015), 433–441  mathnet  crossref  elib
    10. G. G. Braichev, “The exact bounds of lower type magnitude for entire function of order $\rho\in(0,1)$ with zeros of prescribed average densities”, Ufa Math. J., 7:4 (2015), 32–57  mathnet  crossref  isi  elib
    11. O. V. Sherstyukova, “The problem on the minimal type of entire functions of order $\rho\in(0,1)$ with positive zeroes of prescribed densities and step”, Ufa Math. J., 7:4 (2015), 140–148  mathnet  crossref  isi  elib
    12. G. G. Braichev, “The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector”, Sb. Math., 207:2 (2016), 191–225  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. 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”, Ufa Math. J., 8:1 (2016), 108–120  mathnet  crossref  isi  elib
    14. 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  mathnet
    15. 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  mathnet  mathscinet
    16. K. G. Malyutin, T. I. Malyutina, T. V. Shevtsova, “Azarin limiting sets of functions and asymptotic representation of integrals”, Ufa Math. J., 11:2 (2019), 97–113  mathnet  crossref  isi
    17. G. G. Braichev, “On the Lower Indicator of an Entire Function with Roots of Zero Lower Density Lying on a Ray”, Math. Notes, 107:6 (2020), 877–889  mathnet  crossref  crossref  isi  elib
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