Abstract
We study the relationship between the strongly regular growth of an entire function f of order zero, the existence of the angular density of its zeros, the behavior of the Fourier coefficients of the logarithm of f, and the regular growth of the logarithm of the modulus and the argument of f in the L p[0, 2π]-metric, p ≥ 1.
Similar content being viewed by others
References
A. A. Gol’dberg and I. V. Ostrovskii, Distribution of the Values of Meromorphic Functions (Nauka, Moscow, 1970) [in Russian].
B. Ya. Levin, Distribution of the Roots of Entire Functions (Gostekhizdat, Moscow, 1956) [in Russian].
V. S. Azarin, “The regularity of the growth of the Fourier coefficients of the logarithm of the modulus of an entire function,” Funktsional. Anal. Prilozhen. 27, 9–22 (1977).
A. A. Kondratyuk, Fourier series and Meromorphic Functions (Vyshcha Shkola, Lvov, 1988) [in Russian].
A. A. Gol’dberg and N. V. Zabolotskii, “Concentration index of a subharmomic function of zeroth order,” Mat. Zametki 34 (2), 227–236 (1983) [Math. Notes 34 (1–2), 596–601 (1984)].
N. V. Zabolotskii, “Strongly regular growth of entire functions of order zero,” Mat. Zametki 63 (2), 196–208 (1998) [Math. Notes 63 (1–2), 172–182 (1998)].
M. V. Zabolotskii, “An example of entire function strongly regular growth,” Mat. Stud. 13 (2), 145–148 (2000).
N. V. Zabolotskii, “Existence of an angular density of roots of zero-order entire functions,” Mat. Zametki 73 (5), 698–703 (2003) [Math. Notes 73 (5–6), 656–661 (2003)].
E. Seneta, Regularly Varying Functions (Springer-Verlag, Berlin–Heidelberg–New York, 1976; Nauka, Moscow, 1985).
O. Bodnar and M. Zabolotskii, “Distribution of zeros for arguments and Fourier coefficients of the logarithm of an entire function,” Visn. L’viv. Univ., Ser. Mekh.-Mat. 75, 17–27 (2011).
O. V. Bodnar and M. V. Zabolotskii, “Criteria for the regularity of growth of the logarithm of modulus and the argument of an entire function,” Ukrain. Mat. Zh. 62 (7), 885–893 (2010) [Ukrainian Math. J. 62 (7), 1028–1039 (2010)].
R. Z. Kalinets’ and A. A. Kondratyuk, “On the regularity of the growth of the modulus and argument of an entire function in the metric of L [0, 2π] p,” Ukrain. Mat. Zh. 50 (7), 889–896 (1998) [Ukrainian Math. J. 50 (7), 1009–1018 (1998)].
Ya. V. Vasil’kiv, “Asymptotic behavior of the logarithmic derivatives and the logarithms of meromorphic functions of completely regular growth in the L p[0, 2π]. 1,” Mat. Stud. 12 (1), 37–58 (1999).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions (Nauka, Moscow, 1981) [in Russian].
A. V. Bratishchev, “On the inversion of L’Hospital’s rule,” in Mechanics of Solids (Rostov-on-Don, 1985), pp. 28–42 [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N. V. Zabolotskii, O. V. Kostyuk, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 3, pp. 363–374.
Rights and permissions
About this article
Cite this article
Zabolotskii, N.V., Kostyuk, O.V. Regular growth of various characteristics of entire functions of order zero. Math Notes 100, 380–390 (2016). https://doi.org/10.1134/S0001434616090042
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434616090042