On indicator and type of an entire function with roots lying on a ray. (English) Zbl 1502.30079

Summary: A well-known extremal problem is considered: find the exact lower bound for all possible types of entire functions of the order \(\rho\in(1,+\infty)\setminus\mathbb{N}\) with roots on a ray, under the assumption that a lower density of roots is zero, and the upper one takes the given positive value. An approach to this problem is proposed. It based on the study indicator behavior of such entire functions. For the extremal value for any non-integer \(\rho>1\), the best known two-sided estimate is proved. Theoretical statements are supported by the results of numerical calculations.


30D15 Special classes of entire functions of one complex variable and growth estimates
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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[1] Popov, A. Yu., Development of the Valiron-Levin theorem on the least possible type of entire functions with a given upper \(\rho \), J. Math. Sci., 211, 579-616 (2015) · Zbl 1342.30020
[2] Braichev, G. G.; Sherstyukov, V. B., Sharp bounds for asymptotic characteristics of growth of entire functions with zeros on given sets, Fundam. Prikl. Mat., 22, 51-97 (2018) · Zbl 1451.30054
[3] Popov, A. Yu., The least possible type under the order \(\rho<1\) of canonical products with positive zeros of a given upper \(\rho \)-density, Moscow Univ. Math. Bull., 60, 32-36 (2005) · Zbl 1101.30009
[4] 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, 226-239 (2009) · Zbl 1177.30037
[5] Valiron, G., Sur les fonctions entières d’ordre nul et d’ordre fini et en particulier les fonctions à correspondance régulièr, Ann. Fac. Sci. Toulouse, 5, 117-257 (1913) · JFM 46.1462.03
[6] Braichev, G. G.; Sherstyukov, V. B., On the least possible type of entire functions of order \(\rho\in(0,1)\), Izv. Math., 75, 1-27 (2011) · Zbl 1214.30015
[7] Braichev, G. G.; Sherstyukov, V. B., Estimates of indicators of an entire function with negative roots, Vladikavk. Mat. Zh., 22, 30-46 (2020) · Zbl 1474.30189
[8] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Vol. 27 of Encyclopedia of Mathematics and its Applications (Cambridge Univ. Press, Cambridge, 1987). · Zbl 0617.26001
[9] A. Denjoy, ‘‘Sur les produits canoniques d’ordre infini,’’ J. Math. Pures Appl., 6e Ser. 6, 1-136 (1910). · JFM 41.0462.02
[10] Braichev, G. G., On the lower indicator of an entire function with roots of zero lower density lying on a ray, Math. Notes, 107, 877-889 (2020) · Zbl 1442.30029
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