Abstract
This paper provides an overview of the latest research on the two-sided estimates of classical characteristics of growth of entire functions such as the type and the lower type in terms of the ordinary or average densities of the distribution of zeros. We give also accurate estimates of the type of an entire function, taking into account additionally the step and the lacunarity index of the sequence of zeros. The results under consideration are based on the solution of extremal problems in classes of entire functions with restrictions on the behavior of the zero set. Particular attention is paid to the following important cases of the location of zeros: on a ray, on a straight line, on a number of rays, in the angle, or arbitrarily in the complex plane.
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References
A. V. Abanin, “Sampling sets for the space of holomorphic functions of polynomial growth in a ball,” Ufa Math. J., 7, No. 4, 3–14 (2015).
N. I. Ahiezer, Lectures in the Theory of Approximation [in Russian], Nauka, Moscow (1965).
M. I. Andrashko, “The extremal indicator of an entire function with positive zeros of order less than 1,” Dopov. Akad. Nauk Ukr. RSR, 7, 869–872 (1960).
A. E. Avetisyan, “On entire functions of order ρ (1 < ρ < 2),” Sov. J. Contemp. Math. Anal., Arm. Acad. Sci., 23, No. 6, 45–61 (1988).
V. S. Azarin, “On rays of completely regular growth of an entire function,” Sb. Math., 8, No. 4, 437–450 (1969).
V. S. Azarin, “On regularity of growth for functionals on entire functions,” Teor. Funkt. Funkt. Anal. Pril. Kharkov, 16, 109–137 (1972).
V. S. Azarin, “On extremal problems on entire functions,” Teor. Funkt. Funkt. Anal. Pril. Kharkov, 18, 18–50 (1973).
N. K. Bary, A Treatise on Trigonometric Series, Vols. I, II, Pergamon Press, Oxford (1964).
L. Bieberbach, Analytische Fortsetzung, Springer, Berlin (1955).
N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encycl. Math. Its Appl., Vol. 27, Cambridge Univ. Press, Cambridge (1987).
R. P. Boas, Entire Functions, Academic Press, New York (1954).
G. G. Braichev, “Index of lacunarity,” Math. Notes, 53, No. 6, 565–568 (1993).
G. G. Braichev, Introduction in the Theory of Growth of Convex and Entire Functions [in Russian], Prometei, Moscow (2005).
G. G. Braichev, “Exact estimates of types of entire functions of an order ρ ∈ (0, 1) with zeroes on the ray,” Ufimsk. Mat. Zh., 4, No. 1, 29–37 (2012).
G. G. Braichev, “The least type of an entire function of order ρ ∈ (0, 1) having positive zeros with prescribed averaged densities,” Sb. Math., 203, No. 7, 950–975 (2012).
G. G. Braichev, “Exact relationships between certain characteristics of growth for complex sequences,” Ufa Math. J., 5, No. 4, 16–29 (2013).
G. G. Braichev, “Sharp estimates of types of entire functions with zeros on rays,” Math. Notes, 97, No. 4, 510–520 (2015).
G. G. Braichev, “The exact bounds of lower type magnitude for entire function of order ρ ∈ (0, 1) with zeros of prescribed average densities,” Ufa Math. J., 7, No. 4, 32–57 (2015).
G. G. Braichev, “The smallest type of an entire function with the roots of given averaged densities located on the rays or in the angle,” Mat. Sb., 207, No. 2, 45–80 (2016).
G. G. Braichev and O. V. Sherstjukova, “The greatest possible lower type of entire functions of order ρ ∈ (0, 1) with zeros of fixed ρ-densities,” Math. Notes, 90, No. 2, 189–203 (2011).
G. G. Braichev and V. B. Sherstyukov, “Relation between type of an entire function of finite order and the densities of its zeroes,” in: Proc. of XIV Int. Conf. “Mathematics. Economics. Education,” Abrau-Dyurso [in Russian] (2006), pp. 52–55.
G. G. Braichev and V. B. Sherstyukov, “Sharp relation between densities of the zeros of entire functions of finite order,” Mat. Stud., 30, No. 2, 183–188 (2008).
G. G. Braichev and V. B. Sherstyukov, “On the least possible type of entire functions of order ρ ∈ (0, 1) with positive zeros,” Izv. Math., 75, No. 1, 1–27 (2011).
G. G. Braichev and V. B. Sherstyukov, “On the growth of entire functions with discretely measurable zeros,” Math. Notes, 91, No. 5, 630–644 (2012).
A. V. Bratishchev, “A type of lower estimate for entire functions of finite order, and some applications,” Math. USSR Izv., 24, No. 3, 415–438 (1985).
A. Denjoy, “Sur les produits canoniques d’ordre infini,” J. Math. Pures Appl., 6e ser., 6, 1–136 (1910).
A. Eremenko and P. Yuditskii, “An extremal problem for a class of entire functions of exponential type,” arXiv:0807.2054V1[math.CV] (2008).
A. A. Goldberg, A. E. Eremenko, and I. V. Ostrovskii, “On the sum of entire functions of completely regular growth,” Izv. Akad. Nauk Arm. SSR Ser. Mat., 18, No. 1, 3–15 (1983).
A. A. Gol’dberg, “The integral over a semi-additive measure and its application to the theory of entire functions. I,” Mat. Sb., 58 (100), No. 3, 289–334 (1962).
A. A. Gol’dberg, “The integral over a semi-additive measure and its application to the theory of entire functions. II,” Mat. Sb., 61 (103), No. 3, 334–349 (1963).
A. A. Gol’dberg, “The integral over a semi-additive measure and its application to the theory of entire functions. III,” Mat. Sb., 65 (107), No. 3, 414–453 (1964).
A. A. Gol’dberg, “The integral over a semi-additive measure and its application to the theory of entire functions. IV,” Mat. Sb., 66 (108), No. 3, 411–457 (1965).
A. A. Gol’dberg, B. Ya. Levin, and I. V. Ostrovskii, “Entire and meromorphic functions,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 85, 5–185 (1991).
A. A. Gol’dberg and I. V. Ostrovskii, “Indicators of entire functions of finite order that can be represented by Dirichlet series,” Leningr. Math. J., 2, No. 3, 589–612 (1991).
N. V. Govorov, “The extremal indicator of an entire function with positive zeros of given upper and lower density,” Dopov. Akad. Nauk Ukr. RSR, 2, 148–150 (1966).
C. Horovitz, B. Korenblum, and B. Pinchuk, “Sampling sequences for A−∞,” Michigan Math. J., 44, No. 2, 389–398 (1997).
B. N. Khabibullin, “On the type of entire and meromorphic functions,” Mat. Sb. 183, No. 11, 35–44 (1992).
B. N. Khabibullin, “On the growth of entire functions of exponential type near a straight line,” Math. Notes, 70, No. 4, 560–573 (2001).
B. N. Khabibullin, Completeness of Systems of Exponentials and Uniqueness Sets [in Russian], Publ. Centre Bashkir. State Univ., Ufa (2006).
B. N. Khabibullin, “Zero sequences of holomorphic functions, representation of meromorphic functions. II. Entire functions,” Sb. Math., 200, No. 2, 283–312 (2009).
A. A. Kondratjuk, “The extremal indicator for entire functions with positive zeros,” Litovsk. Mat. Sb., 7, No. 1, 79–117 (1967).
A. A. Kondratjuk, “The extremal indicator for entire functions with positive zeros, II,” Litovsk. Mat. Sb., 8, No. 1, 65–85 (1968).
A. A. Kondratyuk, “On the extremal indicator of entire functions with positive zeros,” Sib. Math. J., 11, No. 5, 805–811 (1970).
A. A. Kondratyuk, Fourier Series and Meromorphic Functions [in Russian], Vyshcha Shkola, Lvov (1988).
Yu. F. Korobeinik, Solvability of Some General Classes of Linear Operator Equations in the Complex Domain [in Russian], Rostov State Univ. Publ. House, Rostov-on-Don (2005).
I. F. Krasichkov, “Lower bound for entire functions of finite order,” Sib. Mat. Zh., 6, No. 4, 840–861 (1965).
A. S. Krivosheev, “Regular growth of systems of functions and systems of non-homogeneous convolution equations in convex domains of the complex plane,” Izv. Math., 64, No. 5, 939–1001 (2000).
A. S. Krivosheev, “A criterion for analytic continuation of functions from invariant subspaces in convex domains of the complex plane,” Izv. Math., 68, No. 1, 43–76 (2004).
A. S. Krivosheev and S. N. Gantsev, “On generators in ideals of entire functions of finite order and type on the plane,” Sib. Math. J., 43, No. 5, 843–857 (2002).
A. F. Leont’ev, Exponential Series [in Russian], Nauka, Moscow (1976).
A. F. Leont’ev, Entire Functions. Exponential Series [in Russian], Nauka, Moscow (1983).
B. Ya. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence (1964).
B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc., Providence (1996).
P. Malliavin and L. A. Rubel, “On small entire functions of exponential type with given zeros,” Bull. Soc. Math. Fr., 89, 175–206 (1961).
S. Mandelbrojt, Séries de Dirichlet. Principes et méthodes, Gauthier-Villars, Paris (1969).
A. I. Markushevich, Selected Chapters of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1976).
F. S. Myshakov, “An analog of the Valiron–Goldberg theorem under a restriction condition on the averaged counting function of zeros,” Math. Notes, 96, No. 5, 831–835 (2014).
F. S. Myshakov and A. Yu. Popov, “A refinement of Gol’dberg’s theorem on estimating the type with respect to a proximate order of an entire function of integer order,” Sb. Math., 206, No. 12, 1771–1796 (2015).
A. Pfluger, “Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen. I,” Commun. Math. Helv., 11, 180–213 (1938).
A. Pfluger, “Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Funktionen. II,” Commun. Math. Helv., 12, 25–69 (1939).
A. Yu. Popov, “Completeness of exponential systems with real exponents of a prescribed upper density in spaces of analytic functions,” Moscow Univ. Math. Bull., 54, No. 5, 47–51 (1999).
A. Yu. Popov, Extremal Problems in the Theory of Entire Functions [in Russian], D.Sc. Dissertation, Moscow State University, Moscow (2005).
A. Yu. Popov, “The least possible type under the order ρ < 1 of canonical products with positive zeros of a given upper ρ-density,” Moscow Univ. Math. Bull., 60, No. 1, 32–36 (2005).
A. Yu. Popov, “On the least type of an entire function of order ρ with roots of a given upper ρ-density lying on one ray,” Math. Notes, 85, No. 2, 226–239 (2009).
A. Yu. Popov, “Asymptotic bounds for the modulus of an entire function of noninteger order in terms of a majorant of its zero counting function,” Dokl. Math., 83, No. 2, 188–192 (2011).
A. Yu. Popov, “The most rapid possible growth of the maximum modulus of a canonical product of noninteger order with a prescribed majorant of the counting function of zeros,” Sb. Math., 204, No. 5, 683–725 (2013).
A. Yu. Popov, “Development of the Valiron–Levin theorem on the least possible type of entire functions with a given upper ρ-density of roots,” J. Math. Sci., 211, No. 4, 579–616 (2015).
R. M. Redheffer, “On even entire functions with zeros having a density,” Trans. Am. Math. Soc., 77, 32–61 (1954).
R. M. Redheffer, “Completeness of sets of complex exponentials,” Adv. Math., 24, No. 1, 1–62 (1977).
L. A. Rubel, “Necessary and sufficient conditions for Carlson’s theorem on entire functions,” Trans. Amer. Math. Soc., 83, No. 2, 417–429 (1956).
A. M. Sedletskii, Classes of Analytic Fourier Transforms and Exponential Approximations [in Russian], Fizmatlit, Moscow (2005).
V. B. Sherstyukov, “The minimum value of the type of an entire function of order less than one with zeros of given densities lying in a sector,” in: Proc. of the 17th Int. Saratov Winter School “Modern Problems of Function Theory and Their Applications” [in Russian], Saratov Univ. Publ. House, Saratov (2014), pp. 308–310.
V. B. Sherstyukov, “Minimal value for the type of an entire function of order ρ ∈ (0, 1), whose zeroes lie in angle and have a prescribed density,” Ufa Math. J., 8, No. 1, 113–125 (2016).
O. V. Sherstyukova, “On extremal type of an entire function of order less than unity with zeros of prescribed densities and step,” Ufimsk. Mat. Zh., 4, No. 1, 161–165 (2012).
O. V. Sherstyukova, “On the least type of entire functions of order ρ ∈ (0, 1) with positive zeros,” Izv. Saratov. Univ. (N. S.), Ser. Mat. Mekh. Inform., 15, No. 4, 433–441 (2015).
O. V. Sherstyukova, “Problem on minimal type of entire functions of order ρ ∈ (0, 1) with positive zeroes of prescribed densities and step,” Ufa Math. J., 7, No. 4, 140–148 (2015).
G. Valiron, “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).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 51–97, 2018.
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Braichev, G.G., Sherstyukov, V.B. Sharp Bounds for Asymptotic Characteristics of Growth of Entire Functions with Zeros on Given Sets. J Math Sci 250, 419–453 (2020). https://doi.org/10.1007/s10958-020-05024-4
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DOI: https://doi.org/10.1007/s10958-020-05024-4