The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector
Sbornik. Mathematics, Tome 207 (2016) no. 2, pp. 191-225 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the problem of the least possible type of entire functions whose zeros have fixed upper and lower averaged densities and lie in a given set. In particular, we solve this problem in several important cases: 1) all zeros lie in a sector, 2) all zeros lie between two straight lines; 3) all zeros lie on rays subdividing the complex plane into equal sectors. Bibliography: 15 titles.
Keywords: type of an entire function, upper and lower averaged densities of zeros.
@article{SM_2016_207_2_a1,
     author = {G. G. Braichev},
     title = {The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a~sector},
     journal = {Sbornik. Mathematics},
     pages = {191--225},
     year = {2016},
     volume = {207},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2016_207_2_a1/}
}
TY  - JOUR
AU  - G. G. Braichev
TI  - The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector
JO  - Sbornik. Mathematics
PY  - 2016
SP  - 191
EP  - 225
VL  - 207
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_2016_207_2_a1/
LA  - en
ID  - SM_2016_207_2_a1
ER  - 
%0 Journal Article
%A G. G. Braichev
%T The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector
%J Sbornik. Mathematics
%D 2016
%P 191-225
%V 207
%N 2
%U http://geodesic.mathdoc.fr/item/SM_2016_207_2_a1/
%G en
%F SM_2016_207_2_a1
G. G. Braichev. The least type of an entire function whose zeros have prescribed averaged densities and lie on rays or in a sector. Sbornik. Mathematics, Tome 207 (2016) no. 2, pp. 191-225. http://geodesic.mathdoc.fr/item/SM_2016_207_2_a1/

[1] B. Ja. Levin, Distribution of zeros of entire functions, Amer. Math. Soc., Providence, RI, 1964, viii+493 pp. | MR | MR | Zbl | Zbl

[2] A. Yu. Popov, “Completeness of exponential systems with real exponents of a prescribed under density in spaces of analytic functions”, Mosc. Univ. Math. Bull., 54:5 (1999), 47–51 | MR | Zbl

[3] R. M. Redheffer, “On even entire functions with zeros having a density”, Trans. Amer. Math. Soc., 77 (1954), 32–61 | DOI | MR | Zbl

[4] A. Yu. Popov, “The least possible type under the order $\rho1$ of canonical products with positive zeros of a given upper $\rho$-density”, Mosc. Univ. Math. Bull., 60:1 (2005), 32–36 | MR | Zbl

[5] 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 | DOI | DOI | MR | Zbl

[6] G. G. Braichev, V. B. Sherstyukov, “On the growth of entire functions with discretely measurable zeros”, Math. Notes, 91:5 (2012), 630–644 | DOI | DOI | MR | Zbl

[7] 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 | DOI | DOI | MR | Zbl

[8] G. G. Braichev, “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 | DOI | MR | Zbl

[9] 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”, J. Math. Sci. (N. Y.), 211:4 (2015), 579–616 | DOI

[10] V. B. Sherstyukov, “Minimalnoe znachenie tipa tseloi funktsii poryadka menshe edinitsy s nulyami zadannykh plotnostei, lezhaschimi v ugle”, Materialy 17-i mezhdunarodnoi Saratovskoi zimnei shkoly “Sovremennye problemy teorii funktsii i ikh prilozheniya”, Izd-vo Saratovskogo un-ta, Saratov, 2014, 308–310

[11] G. G. Braichev, “Sharp estimates of types of entire functions with zeros on rays”, Math. Notes, 97:4 (2015), 510–520 | DOI | DOI | MR | Zbl

[12] A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and series, v. 1, Elementary functions, Gordon Breach, New York, 1986, 798 pp. | MR | MR | Zbl | Zbl

[13] G. G. Braichev, Vvedenie v teoriyu rosta vypuklykh i tselykh funktsii, Prometei, M., 2005, 232 pp.

[14] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987, xx+491 pp. | DOI | MR | Zbl

[15] R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954, x+276 pp. | MR | Zbl