Geodesic
Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions
Ufa mathematical journal, Tome 15 (2023) no. 4, pp. 31-41 Cet article a éte moissonné depuis la source Math-Net.Ru

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The idea to write this note arose during the discussion that followed the report of the first author at the International Scientific Conference “Ufa Autumn Mathematical School-2022”. We propose three general methods for constructing uniqueness sets in classes of entire functions with growth restrictions. In all three cases, the sequence of zeros of an entire function with special properties is chosen as such set. The first method is related to Sylvester famous problem on the smallest circle containing a given set of points on a plane, and theorems of convex geometry. The second method initially relies on Helly theorem on the intersection of convex sets and its application to the possibility of covering one set by shifting another. The third method is based on the classical Jensen formula, which allows one to estimate the type of an entire function via the averaged upper density of the sequence of its zeros. We present only basic results now. The development of our approaches is expected to be presented in subsequent works.
Keywords: Sylvester problem, Jung theorem, Helly theorem, uniqueness set, type of entire function, sequence of zeros, indicator of an entire function, averaged upper density, indicator diagram, smallest circle.
Mots-clés : Jensen formula 
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G. G. Braichev; B. N. Khabibullin; V. B. Sherstyukov. Sylvester problem, coverings by shifts, and uniqueness theorems for entire functions. Ufa mathematical journal, Tome 15 (2023) no. 4, pp. 31-41. http://geodesic.mathdoc.fr/item/UFA_2023_15_4_a2/

[1] J.J. Sylvester, “A question in the geometry of situation”, Quarterly Journal of Mathematics, 1 (1857), 79 | MR

[2] A.R. Alimov, I.G. Tsar'kov, “Chebyshev centres, Jung constants, and their applications”, Russ. Math. Surv., 74:5 (2019), 775–849 | DOI | DOI | MR | Zbl

[3] H. Jung, “Ueber die kleinste Kugel, die eine räumliche Figur einschliesst”, Journal für die reine und angewandte Mathematik, 123 (1901), 241–257 | MR

[4] H. Rademacher, O. Toeplitz, Numbers and figures. Experiments in mathematical thinking, Princeton Univ. Press, Princeton, NJ, 1994 | MR

[5] V.Yu. Protasov, “Helly theorem and around”, Kvant, 3 (2009), 8–14 (in Russian)

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

[7] B.N. Khabibullin, Completeness of exponential systems and uniqueness sets, 4th ed. Bashkir State Univ. Publ., Ufa, 2021 (in Russian)

[8] Yu.F. Korobeĭnik, “Maximal and $\gamma$-sufficient sets. Applications to entire functions. I”, Teor. Funkts., Funkts. Anal. Prilozh., 54, 1990, 42–49 (in Russian)

[9] Yu.F. Korobejnik, “Maximal and $\gamma$-sufficient sets. Applications to entire functions. II”, J. Sov. Math., 59:1 (1992), 599–606 | DOI | MR | MR

[10] F. Carlson, “Über ganzwertige Funktionen”, Mathematishe Zeitschrift, 11 (1921), 1–23 | DOI | MR

[11] A.F. Grishin, K.G. Malyutin, Trigonometrically convex functions, South-West Univ. Publ., Kursk, 2015 (in Russian)

[12] V. Azarin, Growth theory of subharmonic functions, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009 | MR | Zbl

[13] B.N. Khabibullin, “Sequences of non-uniqueness for weight spaces of holomorphic functions”, Russ. Math. (Iz. VUZ), 59:4 (2015), 63–70 | DOI | MR | Zbl

[14] B.N. Khabibullin, F.B. Khabibullin, “On the distribution of zero sets of holomorphic functions. III: Converse theorems”, Funct. Anal. Appl., 53:2 (2019), 110–123 | DOI | DOI | MR | Zbl

[15] B.N. Khabibullin, F.B. Khabibullin, “Necessary and Sufficient Conditions for Zero Subsets of Holomorphic Functions with Upper Constraints in Planar Domains”, Lobachevskii Jour. of Math., 42:4 (2021), 800–810 | DOI | MR | Zbl

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

[17] G.G. Braichev, Extremal problems in theory of relative growth of convex and entire functions, Habilit. Ths., RUDN Univ., M., 2018 (in Russian)

[18] L.A. Rubel, B.A. Taylor, “A Fourier series method for meromorphic and entire functions”, Bulletin de la S.M.F., 96 (1968), 53–96 | MR | Zbl

[19] G.G. Braichev, V.B. Sherstyukov, “Sharp bounds for asymptotic characteristics of growth of entire functions with zeros on given sets”, J. Math. Sci., 250:3 (2020), 419–453 | DOI | MR | Zbl

[20] G.G. Braichev, “On the connection between the growth of zeros and the decrease of Taylor coefficients of entire functions”, Math. Notes, 113:1 (2023), 27–38 | DOI | DOI | MR | Zbl

[21] B.N. Khabibullin, “Helly's theorem and shifts of sets. II. Support function, exponential systems, entire functions”, Ufa Math. J., 6:4 (2014), 122–134 | DOI | MR | Zbl