Estimates of indicators of an entire function with negative roots

G. G. Braichev   ; V. B. Sherstyukov

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 3, pp. 30-46

Résumé

The article continues the series of works by the authors devoted to the study of the relationship between the laws growth of an entire function and the features of the distribution of its roots. The asymptotic behavior of an entire function of finite non-integer order with a sequence of negative roots having the prescribed lower and upper densities is investigated. Particular attention is paid to the case when the sequence of roots has zero lower density. Accurate estimates for the indicator and lower indicator of such a function are given. The angles on the complex plane in which these characteristics are identically equal to zero are described. In some special cases explicit formulas for indicators are proved. Terms used, usual root sequence densities, are simple and illustrative, in contrast to many complicated integral constructions including root counting function that are typical for the growth theory of entire functions. The results are applied to the well-known problem of the extremal type of an entire function of order $\rho\in(0,+\infty)\setminus\mathbb{N}$ with zeros on a ray. This problem has been studied in detail only in the case of $\rho\in(0,1)$. For $\rho>1$, the exact formula for calculating the smallest possible type of such a function in terms of the densities of its roots is still unknown. For the mentioned extreme value, a new two-sided estimate is found that strengthens Popov's results (2009). The conjecture regarding the behavior of the extremal type for $\rho\rightarrow p\in\mathbb{N}$ is formulated.The presentation is supplemented with a brief survey of classical results of Valiron, Levin, Goldberg and recent advances from the works of Popov and of the authors. Some problems on the topic under discussion are outlined.

Bibliographie

[1] Korobeinik Yu. F., Selected Works, v 4-kh tomakh, SMI VSC RAS, Vladikavkaz, 2011–2014 (in Russian)

[2] E. Lindelöf, “Mémoire sur la théorie des fonctions entières de genre fini”, Acta Soc. Sci. Fennicae, 31:1 (1902), 1–79 | MR

[3] G. Valiron, “Sur les fonctions entières d'ordre nul et d'ordre fini et en particulier les fonctions à correspondence régulièr”, Annales de la Faculté des Scinces de Toulouse: Mathématiques, Sér. 3, 5 (1913), 117–257 | DOI | MR

[4] Levin B. Ya., Distribution of Zeros of Entire Functions, Gostekhizdat, M., 1956 (in Russian)

[5] R. P. Boas, Entire Functions, Acad. Press, N.Y., 1954 | MR | Zbl

[6] G. G. Braichev, V. B. Sherstyukov, “Sharp Bounds for Asymptotic Characteristics of Growth of Entire Functions with Zeros on Given Set”, Fundamentalnaya i Prikladnaya Matematika, 22:1 (2018), 51–97 (in Russian)

[7] 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, 75:1 (2011), 1–27 | DOI | DOI | MR | Zbl

[8] Popov A. Yu., “The Least Possible Type under the Order $\rho 1$ of Canonical Products with Positive Zeros of a Given Upper $\rho$-density”, Vestnik Moskovskogo Universiteta Seriya 1. Matematika. Mekhanika, 2005, no. 1, 31–36 (in Russian) | Zbl

[9] Sherstyukov V. B., “Minimal Value for the Type of an Entire Function of Order $\rho\in (0, 1)$, whose Zeros Lie in an Angle and Have a Prescribed Density”, Ufa Mathematical Journal, 8:1 (2016), 108–120 | DOI | MR

[10] Popov A. Yu., “Development of the Valiron-Levin Theorem on the Least Possible Type of Entire Functions with a Given Upper $\rho$-Density of Roots”, Journal of Mathematical Sciences, 211:4 (2015), 579–616 | DOI | MR | Zbl

[11] Braichev G. G., “On the Lower Indicator of an Entire Function with Roots of Zero Lower Density Lying on a Ray”, Mathematical Notes, 107:6 (2020), 907–919 | DOI | DOI | MR | Zbl

[12] Gol'dberg A. A., Levin B. Ya., Ostrovskiy I. V., “Entire and Meromorphic Functions”, Itogi Nauki i Tekhniki. Seriya «Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya», 85, VINITI, M., 1991, 5–185 (in Russian)

[13] Malyutin K. G., Kabanko M. V., Malyutina T. I., “Integrals and Indicators of Subharmonic Functions, I”, Chebyshevskii Sbornik, 19:2 (2018), 272–303 (in Russian) | DOI | Zbl

[14] Malyutin K. G., Kabanko M. V., Malyutina T. I., “Integrals and Indicators of Subharmonic Functions, II”, Chebyshevskii Sbornik, 20:4 (2019), 236–269 (in Russian) | DOI | MR | Zbl

[15] Azarin V. S., “Example of an Entire Function with Given Indicator and Lower Indicator”, Mathematics of the USSR-Sbornik, 18:4 (1972), 541–558 | DOI | MR | Zbl

[16] Azarin V. S., “Indicators of an Entire Function and the Regularity of the Growth of the Fourier Coefficients of the Logarithm of its Modulus”, Functional Analysis and its Applications, 9:1 (1975), 41–42 | DOI | MR | Zbl

[17] N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia Math. Appl., 27, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[18] Kondratyuk A. A., Fridman A. N., “Predel'noe Znachenie Nizhnego Indikatora i Otsenki Snizu dlya Tselykh Funktsiy s Polozhitel'nymi Nulyami”, Ukrainian Mathematical Journal, 24:4 (1972), 488–494 (in Russian)

[19] Kondratyuk A. A., Fridman A. N., “O Nizhnem Indikatore Tseloy Funktsii Nulevogo Roda s Polozhitel'nymi Nulyami”, Ukrainian Mathematical Journal, 24:1 (1972), 106–109 (in Russian) | MR | Zbl

[20] 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”, Mathematical Notes, 85:2 (2009), 226–239 | DOI | DOI | MR | Zbl

[21] A. Denjoy, “Sur les produits canoniques d'ordre infini”, J. Math. Pures Appl. 6e ser., 6 (1910), 1–136