Sharp bounds for asymptotic characteristics of growth of entire functions with zeros on given sets

G. G. Braichev ; V. B. Sherstyukov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2018) no. 1, pp. 51-97.

Résumé

The 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 the 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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