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Mat. Zametki, 2009, Volume 85, Issue 2, Pages 246–260 (Mi mz4645)  

This article is cited in 8 scientific papers (total in 8 papers)

On the Least Type of an Entire Function of Order $\rho$ with Roots of a Given Upper $\rho$-Density Lying on One Ray

A. Yu. Popov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It is well known that the least possible type from the class of entire functions of prescribed order $\rho$ with upper root density 1 (for the exponent $\rho$) is $1/(e\rho)$. The author has proved that if all the roots of entire functions lie on one ray, then the situation is different: the least type for such a class on the set of orders $(1,+\infty)\setminus\mathbb N$ is distinct from zero and is bounded above.

Keywords: entire function, least type of an entire function, upper density of a sequence, Lindelöf theorem

DOI: https://doi.org/10.4213/mzm4645

Full text: PDF file (499 kB)
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English version:
Mathematical Notes, 2009, 85:2, 226–239

Bibliographic databases:

UDC: 517.547.22
Received: 20.03.2008

Citation: A. Yu. Popov, “On the Least Type of an Entire Function of Order $\rho$ with Roots of a Given Upper $\rho$-Density Lying on One Ray”, Mat. Zametki, 85:2 (2009), 246–260; Math. Notes, 85:2 (2009), 226–239

Citation in format AMSBIB
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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. G. G. Braichev, V. B. Sherstyukov, “On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros”, Izv. Math., 75:1 (2011), 1–27  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    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. 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
    4. 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
    5. 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
    6. 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
    7. 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
    8. 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
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