Geodesic
On least type of entire function with given subsequence of zeros
Ufa mathematical journal, Tome 14 (2022) no. 3, pp. 17-21 Cet article a éte moissonné depuis la source Math-Net.Ru

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This note is based on the authors' report on International Scientific Conference “Ufa Autumn Mathematical School – 2021”. We discuss the following problem. Let we are given a non-integer number $\rho>0$ and a sequence of complex numbers $\Lambda$ having a finite upper $\rho$-density. Then, as it is known by the classical Lindelöf theorem, there exists a (not identically zero) entire function $f$ of a finite type of the order $\rho$, for which $\Lambda$ is a sequence of all its zeroes. The question is how much can the type of such function change if, apart of the elements in $\Lambda$, it can have other zeroes of an arbitrary multiplicity. We show the possibilities of applying one general theorem proved by B.N. Khabibullin in 2009. In order to do this we use recent results containing the exact formulae for calculating extremal type in classes of entire functions with various restrictions on the distribution of zeroes. The case of entire $\rho$ possesses certain features and in this work we almost not consider it.
Keywords: entire function, sequence of zeroes, subsequence of zeroes, type of entire function, extremal problem. 
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G. G. Braichev; O. V. Sherstyukova. On least type of entire function with given subsequence of zeros. Ufa mathematical journal, Tome 14 (2022) no. 3, pp. 17-21. http://geodesic.mathdoc.fr/item/UFA_2022_14_3_a1/

[1] R.P. Boas., Entire Functions, Academic Press Inc., New York, 1954 | MR | Zbl

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

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

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

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

[6] O.V. Sherstyukova, “The problem on the minimal type of entire functions of order $\rho\in(0,\,1)$ with positive zeroes of prescribed densities and step”, Ufa Math. J., 7:4 (2015), 140–148 | DOI | MR | Zbl

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

[8] 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

[9] G.G. Braichev, V.B. Sherstyukov, “On Indicator and Type of an Entire Function with Roots Lying on a Ray”, Lobachevskii Journal of Math., 43:3 (2022), 1288–1298 | DOI | MR