Geodesic
On the Lower Indicator of an Entire Function
Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 817-832.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper deals with an entire function of noninteger order with a sequence of negative roots having (for this order) zero lower and finite upper densities. Sharp estimates for the lower indicator of such a function are obtained. It is proved that, in some angles, this characteristic is identically zero, and its form in the other angles is obtained provided that the sequence of roots of the entire function sufficiently rapidly tends to infinity.
Keywords: entire function, type and lower type of a function, indicator and lower indicator, upper and lower densities of a set of roots, completely regular growth. 
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G. G. Braichev. On the Lower Indicator of an Entire Function. Matematičeskie zametki, Tome 107 (2020) no. 6, pp. 817-832. http://geodesic.mathdoc.fr/item/MZM_2020_107_6_a1/

[1] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, M., 1956 | MR | Zbl

[2] B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc., Providence, RI, 1996 | MR | Zbl

[3] 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

[4] A. Yu. Popov, “Naimenshii vozmozhnyi tip pri poryadke $\rho1$ kanonicheskikh proizvedenii s polozhitelnymi nulyami zadannoi verkhnei $\rho$-plotnosti”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2005, no. 1, 31–36 | MR | Zbl

[5] G. G. Braichev, V. B. Sherstyukov, “O naimenshem vozmozhnom tipe tselykh funktsii poryadka $\rho\in(0,1)$ s polozhitelnymi nulyami”, Izv. RAN. Ser. matem., 75:1 (2011), 3–28 | DOI | MR | Zbl

[6] A. Yu. Popov, “O naimenshem tipe tseloi funktsii poryadka $\rho$ s kornyami zadannoi verkhnei $\rho$-plotnosti, lezhaschimi na odnom luche”, Matem. zametki, 85:2 (2009), 246–260 | DOI | MR | Zbl

[7] O. V. Sherstyukova, “Zadacha o naimenshem tipe tselykh funktsii poryadka $\rho\in(0,1)$ s polozhitelnymi nulyami zadannykh plotnostei i shaga”, Ufimsk. matem. zhurn., 7:4 (2015), 146–154

[8] G. Valiron, “Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondance régulière”, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3), 5 (1913), 117–257 | DOI | MR

[9] G. G. Braichev, O. V. Sherstyukova, “Naibolshii vozmozhnyi nizhnii tip tseloi funktsii poryadka $\rho\in(0;1)$ s nulyami fiksirovannykh $\rho$-plotnostei”, Matem. zametki, 90:2 (2011), 199–215 | DOI | MR | Zbl

[10] A. F. Leontev, Tselye funktsii. Ryady eksponent, Nauka, M., 1976 | MR

[11] A. Pfluger, “Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen. I”, Comment. Math. Helv., 11 (1938), 180–213 | DOI | MR

[12] A. Pfluger, “Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen. II”, Comment. Math. Helv., 12 (1939), 25–69 | DOI | MR

[13] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[14] A. A. Goldberg, “Integral po poluadditivnoi mere i ego prilozhenie k teorii tselykh funktsii. IV”, Matem. sb., 66 (108):3 (1965), 411–457 | MR | Zbl

[15] A. V. Abanin, V. V. Yudelevich, “Ob obraschenii teoremy Shtoltsa”, Izv. vuzov. Sev.-Kavk. region. Estestv. nauki., 2016, no. 2, 5–9