Geodesic
The Sylvester problem and uniqueness sets in classes of entire functions
Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 25-37.

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

In this paper, we study the problem of finding, by a chosen sequence of complex numbers tending to infinity, the widest possible class of entire functions in a given scale for which this sequence is a uniqueness set. Within the framework of this general problem, we establish uniqueness theorems in various classes of entire functions, distinguished by restrictions on the type and indicator under a refined order. In particular, we complement the previously proven uniqueness theorem, using the concept of the Sylvester circle of the indicator diagram of an entire function of exponential type. We discuss the accuracy of the results obtained and their connection with known facts.
Keywords: Sylvester circle, indicator diagram, entire functions, uniqueness set. 
@article{CMFD_2024_70_1_a2,
     author = {G. G. Braichev},
     title = {The {Sylvester} problem and uniqueness sets in classes of entire functions},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {25--37},
     publisher = {mathdoc},
     volume = {70},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a2/}
}
TY  - JOUR
AU  - G. G. Braichev
TI  - The Sylvester problem and uniqueness sets in classes of entire functions
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2024
SP  - 25
EP  - 37
VL  - 70
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a2/
LA  - ru
ID  - CMFD_2024_70_1_a2
ER  - 
%0 Journal Article
%A G. G. Braichev
%T The Sylvester problem and uniqueness sets in classes of entire functions
%J Contemporary Mathematics. Fundamental Directions
%D 2024
%P 25-37
%V 70
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a2/
%G ru
%F CMFD_2024_70_1_a2
G. G. Braichev. The Sylvester problem and uniqueness sets in classes of entire functions. Contemporary Mathematics. Fundamental Directions, Functional spaces. Differential operators. Problems of mathematics education, Tome 70 (2024) no. 1, pp. 25-37. http://geodesic.mathdoc.fr/item/CMFD_2024_70_1_a2/

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

[2] Braichev G. G., Ekstremalnye zadachi v teorii vypuklykh i tselykh funktsii, Diss. dokt. fiz.-mat. nauk, RUDN, M., 2018

[3] Braichev G. G., “O svyazi mezhdu rostom nulei i ubyvaniem teilorovskikh koeffitsientov tseloi funktsii”, Mat. zametki, 113:1 (2023), 32–45 | DOI | MR | Zbl

[4] Braichev G. G., Khabibullin B. N., Sherstyukov V. B., “Zadacha Silvestra, pokrytiya sdvigami i teoremy edinstvennosti dlya tselykh funktsii”, Ufimskii mat. zh., 15:4 (2023), 30–41

[5] Braichev G. G., Sherstyukov V. B., “Tochnye otsenki asimptoticheskikh kharakteristik rosta tselykh funktsii s nulyami na zadannykh mnozhestvakh”, Fundam. i prikl. mat., 22:1 (2018), 51–97

[6] Braichev G. G., Sherstyukova O. V., “O naimenshem tipe tseloi funktsii s zadannoi podposledovatelnostyu nulei”, Ufimskii mat. zh., 14:3 (2022), 17–22 | MR | Zbl

[7] Grishin A. F., Van Kuin N., “Tselye funktsii s napered zadannym nulevym utochnennym poryadkom”, Zap. nauch. sem. POMI, 424, 2014, 141–153

[8] Levin B. Ya., Raspredelenie kornei tselykh funktsii, GITTL, M., 1956

[9] Popov A. Yu., “Razvitie teoremy Valirona—Levina o naimenshem vozmozhnom tipe tseloi funktsii s zadannoi verkhnei $\rho$-plotnostyu kornei”, Sovrem. mat. Fundam. napravl., 49, 2013, 132–164

[10] Filevich P. V., “Indikator tselykh funktsii s silno koleblyuschimisya koeffitsientami”, Mat. stud., 35:2 (2011), 142–148 | MR | Zbl

[11] Khabibullin B. N., “O tipe tselykh i meromorfnykh funktsii”, Mat. sb., 183:11 (1992), 35–44 | Zbl

[12] Khabibullin B. N., “Posledovatelnost nulei golomorfnykh funktsii, predstavlenie meromorfnykh funktsii. II. Tselye funktsii”, Mat. sb., 200:2 (2009), 129–158 | DOI | Zbl

[13] Khabibullin B. N., Polnota sistem eksponent i mnozhestva edinstvennosti, RITs BashGU, Ufa, 2012

[14] Sherstyukov V. B., Asimptoticheskie svoistva tselykh funktsii, korni kotorykh lezhat v nekotorom ugle, Diss. dokt. fiz.-mat. nauk, MGU, M., 2016

[15] Earl E. P., “Note in the constraction of proximate orders”, J. London Math. Soc., 43 (1968), 695–698 | DOI | MR | Zbl

[16] Earl E. P., Hayman W. K., “Smooth majorants for functions of arbitrarily rapid grouth”, Math. Proc. Camb. Phil. Soc., 109:3 (1991), 565–569 | DOI | MR | Zbl

[17] Valiron G., “Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondance règulière”, Ann. Fac. Sci. Toulouse Math. (3), 1913, no. 5, 117–257 | DOI | MR