Geodesic
On the least type of entire functions of order $\rho\in(0,1)$ with positive zeros
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 433-441 
O. V. Sherstyukova. On the least type of entire functions of order $\rho\in(0,1)$ with positive zeros. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 433-441. http://geodesic.mathdoc.fr/item/ISU_2015_15_4_a7/
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     title = {On the least type of entire functions of order $\rho\in(0,1)$ with positive zeros},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the theory of extremal problems in classes of entire functions with constraints on the growth and distribution of zeros and is associated with problems of completeness of exponential systems in the complex domain. The question of finding the exact lower bound for types of all entire functions of order $\rho\in(0,1)$ whose zeros lie on the ray and have prescribed upper $\rho$-density and $\rho$-step is discussed. It is shown that the infimum is attained in this problem, and a detailed construction of the extremal function is given. This result gives a complete solution of the extremal problem and generalizes preceding result of A. Yu. Popov.

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

[2] Popov A. Yu., “On the least type of an entire function of order $\rho$ with roots of a given upper $\rho$-density lying on one ray”, Math. Notes, 85:1–2 (2009), 226–239 | DOI | DOI | MR | Zbl

[3] Braichev G. G., Sherstyukov V. B., “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

[4] Sherstyukova O. V., “On the influence of the step sequence of zeros of entire functions of order less than one on the value of its type”, Science in high schools : mathematics, computer science, physics, education, Moscow Pedagogical State University, M., 2010, 192–195 (in Russian)

[5] Braichev G. G., Sherstyukova O. V., “The greatest possible lower type of entire functions of order $\rho\in(0,\,1)$ with zeros of fixed $\rho$-densities”, Math. Notes, 90:1–2 (2011), 189–203 | DOI | DOI | MR | Zbl

[6] Sherstyukova O. V., “On extremal type of an entire function of order less than unity with zeros of prescribed densities and step”, Ufa Mathematical Journal, 4:1 (2012), 151–155 | MR | Zbl

[7] Braichev G. G., “The least type of an entire function of order $\rho\in(0,1)$ having positive zeros with prescribed averaged densities”, Sbornik: Mathematics, 203:7 (2012), 950–975 | DOI | DOI | MR | Zbl

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

[9] Valiron G., “Sur les fonctions entieres d'ordre nul et d'ordre fini et en particulier les fonctions a correspondance reguliere”, Annales de la faculte des sciences de Toulouse. Ser. 3, 5 (1913), 117–257 | DOI | MR