Abstract
We establish a necessary and sufficient condition for the coefficients a n of an entire function \(f(z) = \sum {_{n = 0}^\infty } {\text{ }}a_n z^n \) under which its central index and the logarithms of the maximum of the modulus and the maximum term are regularly varying functions. We construct an entire function the logarithm of the maximum of whose modulus is a regularly varying function, whereas the Nevanlinna characteristic function is not a regularly varying function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).
N. V. Zabolotskii and M. N. Sheremeta, “On the slow growth of main characteristics of entire functions,” Mat. Zametki, 65, No. 2, 206-214 (1999).
O. B. Skaskiv and O. M. Trakalo, “On the slow growth of the counting function of an additional sequence,” Visn. Lviv. Nats. Univ. Ser. Mekh.-Mat., Issue 57, 36-40 (2000).
G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Springer, Berlin (1964).
E. F. Shchuchinskaya, “On the Borel inequality for entire functions of finite order,” Izv. SKNTsVSh. Estest. Nauk., No. 1, 22-23 (1981).
“Research problems,” in: Mathematics: Collection of Translations of Foreign Works [in Russian], No. 7:5, Izd. Inostr. Lit., Moscow (1963), pp. 133-136.
A. A. Gol'dberg, “Three examples of entire functions,” Dopov. Akad. Nauk Ukr. SSR, No. 4, 443-446 (1963).
A. A. Gol'dberg and I. V. Ostrovskii, Distribution of Values of Meromorphic Functions [in Russian], Nauka, Moscow (1970).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Filevych, P.V., Sheremeta, M.M. On the Regular Variation of Main Characteristics of an Entire Function. Ukrainian Mathematical Journal 55, 1012–1024 (2003). https://doi.org/10.1023/B:UKMA.0000010600.46493.2c
Issue Date:
DOI: https://doi.org/10.1023/B:UKMA.0000010600.46493.2c