Geodesic
On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros
Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 1-27.

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

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. 
@article{IM2_2011_75_1_a0,
     author = {G. G. Braichev and V. B. Sherstyukov},
     title = {On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros},
     journal = {Izvestiya. Mathematics },
     pages = {1--27},
     publisher = {mathdoc},
     volume = {75},
     number = {1},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a0/}
}
TY  - JOUR
AU  - G. G. Braichev
AU  - V. B. Sherstyukov
TI  - On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros
JO  - Izvestiya. Mathematics 
PY  - 2011
SP  - 1
EP  - 27
VL  - 75
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a0/
LA  - en
ID  - IM2_2011_75_1_a0
ER  - 
%0 Journal Article
%A G. G. Braichev
%A V. B. Sherstyukov
%T On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros
%J Izvestiya. Mathematics 
%D 2011
%P 1-27
%V 75
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a0/
%G en
%F IM2_2011_75_1_a0
G. G. Braichev; V. B. Sherstyukov. On the least possible type of entire functions of order $\rho\in(0,1)$ with positive zeros. Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 1-27. http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a0/

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

[2] 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

[3] N. V. Govorov, “Ekstremalnii ïndikator tsiloï funktsiï z dodatnimi nulyami zadanoï verkhnoï ta nizhnoï gustini”, Dop. AN URSR, 2 (1966), 148–150 | MR | Zbl

[4] A. A. Kondratyuk, “On the extremal indicator of entire functions with positive zeros”, Siberian Math. J., 11:5 (1970), 805–811 | DOI | MR | Zbl | Zbl

[5] M. I. Andrashko, “Ekstremalnii indikator tsiloï funktsiï z dodatnimi nulyami poryadka menshe odinitsi”, Dop. AN URSR, 7 (1960), 869–872 | Zbl

[6] B. N. Khabibullin, “Zero sequences of holomorphic functions, representation of meromorphic functions. II. Entire functions”, Sb. Math., 200:2 (2009), 283–312 | DOI | MR | Zbl

[7] 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”, Math. Notes, 85:1–2 (2009), 226–239 | DOI | MR | Zbl

[8] A. Yu. Popov, Ekstremalnye zadachi v teorii tselykh funktsii, Dis. ... dokt. fiz.-matem. nauk, MGU, M., 2005

[9] A. Yu. Popov, “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

[10] R. Ph. Boas, jr., Entire functions, Academic Press, New-York, 1954 | MR | Zbl

[11] A. Yu. Popov, “Exact estimates for the condensation index of a sequence of positive numbers”, Proc. Steklov Inst. Math., Suppl. 1, 2004, Topology, mathematical control theory and differential equations, approximation theory (2004), S183–S206 | MR | Zbl

[12] G. G. Braichev, V. B. Sherstyukov, “On an extremal problem related to the completeness of a system of exponentials in the disk”, Asian-Eur. J. Math., 1:1 (2008), 15–26 | DOI | MR | Zbl

[13] R. C. Buck, “On the distribution of the zeros of an entire function”, J. Indian Math. Soc. (N.S.), 16:4 (1952), 147–149 | MR | Zbl

[14] G. G. Braichev, V. B. Sherstyukov, “Sharp relation between densities of the zeros of entire functions of finite order”, Mat. Stud., 30:2 (2008), 183–188 | MR | Zbl

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

[16] G. G. Braichev, V. B. Sherstyukov, “Asimptoticheskaya otsenka ekstremalnoi velichiny v odnoi zadache teorii tselykh funktsii”, Tez. dokladov Mezhdunarodnoi nauchno-obrazovatelnoi konferentsii “Nauka v vuzakh: matematika, fizika, informatika. Problemy vysshego i srednego professionalnogo obrazovaniya”, RUDN, M., 2009, 241–252

[17] B. N. Khabibullin, Polnota sistem eksponent i mnozhestva edinstvennosti, RITs BashGU, Ufa, 2006

[18] A. F. Leontev, Ryady eksponent, Nauka, M., 1976 | MR | Zbl

[19] P. Malliavin, L. A. Rubel, “On small entire functions of exponential type with given zeros”, Bull. Soc. Math. France, 89 (1961), 175–206 | MR | Zbl

[20] B. N. Khabibullin, “On the type of entire and meromorphic functions”, Russian Acad. Sci. Sb. Math., 77:2 (1994), 293–301 | DOI | MR | Zbl

[21] L. A. Rubel, “Necessary and sufficient conditions for Carlson's theorem on entire functions”, Trans. Amer. Math. Soc., 83:2 (1956), 417–429 | DOI | MR | Zbl

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