Sharp Bounds for Asymptotic Characteristics of Growth of Entire Functions with Zeros on Given Sets
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SHARP BOUNDS FOR ASYMPTOTIC CHARACTERISTICS OF GROWTH OF ENTIRE FUNCTIONS WITH ZEROS ON GIVEN SETS UDC 517.547.22
G. G. Braichev and V. B. Sherstyukov
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.
1. Main Definitions and Preliminaries In several branches of complex analysis, special emphasis is placed on the relation between the asymptotic behavior of a function and the distribution of its zeros. In the framework of the theory of entire functions of completely regular growth, such a dependence was already developed in the mid-twentieth century (see [33, 52, 53, 59, 60]) with abundant applications (see, for example, [2, 45, 51, 71]). For a further development of this classical theory, see, for example, [10, 44]. The present need in the intensive study of entire functions of not completely regular growth arises from new demands in the theory of interpolation, approximation, and analytic continuation in complex domains, problems of the spectral theory of operators and probability theory. The present paper gives a survey of recent exact results on the growth of entire functions of finite order whose zeros lie on a fixed set and have prescribed density characteristic. These results are formulated in terms of extremal problems in the corresponding classes of entire functions. Important results in the genesis and development of this field are due to Valiron [77], Levin (see, for example, [52, Chap. V, Sec. 5, Theorem 12]), Redheffer [68], and Gol’dberg [29–32] and later by Govorov [35], Kondratyuk [41–43], Andrashko [3], and Khabibullin [37, 40]. We give necessary definitions and notation. The type and lower type of order ρ > 0 are defined, respectively, as ln max |f (z)| σρ (f ) = lim
r→+∞
|z|=r rρ
ln max |f (z)| ,
σ ρ (f ) = lim
r→+∞
|z|=r rρ
.
These quantities are the traditional characteristics of the global growth of an entire function f (z). Briefly (with indication of the dependence on ρ) these quantities are called the ρ-type and the lower ρ-type. A function f (z) is said to be of perfectly regular growth of modulus if there exists the limit ln max |f (z)| lim
r→+∞
|z|=r rρ
= σρ (f ) = σ ρ (f ).
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 22, No. 1, pp. 51–97, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2503–0419
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The growth of an entire function
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