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

Braichev, G. G.; Sherstyukov, V. B.

Lobachevskii J. Math. 43, No. 3, 539-549 (2022).

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.

MSC:

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)

Keywords:

entire function of finite order; roots on a ray

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