We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of rays in terms of Fourier coefficients of their logarithmic derivative.

[1] Gol'dberg A.A. B.Ya. Levin is a creator of the theory of entire functions of completely regular growth. Mat. Fiz., Anal., Geom. 1994, 1 (2), 186-192. (in Russian)

[2]  Gol'dberg A.A., Korenkov N.E. Asymptotic behavior of logarithmic derivative of entire function of completely regular growth. Sib. Math. J. 1980, 21 (3), 363-375. doi:10.1007/BF00968180 (translation of Sib. Mat. Zh. 1980, 21 (3), 63-79. (in Russian))

[3] Gol'dberg A.A., Levin B.Ya., Ostrovskii I.V. Entire and meromorphic functions. In: VINITI Series in Contemporary Problems of Mathematics, Fundamental Trends, 85. VINITI, Moscow 1991, pp. 5-186. (in Russian)

[4] Gol'dberg A.A., Sodin M.L., Strochik N.N. Meromorphic functions of completely regular growth and their logarithmic derivatives. Sib. Math. J. 1992, 33 (1), 34-40. doi:10.1007/BF00972934 (translation of Sib. Mat. Zh. 1992, 33 (1), 44-52. (in Russian))

[5] Gol'dberg A.A., Strochik N.N. Asymptotic behavior of meromorphic functions of completely regular growth and of their logarithmic derivatives. Sib. Math. J. 1985, 26 (6), 802-809. doi:10.1007/BF00969100 (translation of Sib. Mat. Zh. 1985, 26 (6), 29-38. (in Russian))

[6] Hirnyk M.O. Subharmonic functions of improved regular growth. Dopov. Nats. Akad. Nauk Ukr. 2009, 4, 13-18. doi:10.1007/s11253-012-0624-2 (in Ukrainian)

[7] Khats' R.V. On entire functions of improved regular growth of integer order with zeros on a finite system of rays. Mat. Stud. 2006, 26 (1), 17-24.

[8] Khats' R.V. Regularity of growth of Fourier coefficients of entire functions of improved regular growth. Ukr. Math. J. 2012, 63 (12), 1953-1960. doi:10.1007/s11253-012-0624-2 (translation of Ukr. Mat. Zh. 2011, 63 (12), 1717-1723. (in Ukrainian))

[9] Khats' R.V. Asymptotic behavior of averaging of entire functions of improved regular growth. Carpathian Math. Publ. 2013, 5 (1), 129-133. doi:10.15330/cmp.5.1.129-133

[10] Khats' R.V. Asymptotic behavior of entire functions of improved regular growth in the metric of $L^p[0, 2π]$. Carpathian Math. Publ. 2013, 5 (2), 341-344. doi:10.15330/cmp.5.2.341-344

[11] Kondratyuk A.A. Fourier series and meromorphic functions. Vyshcha shkola, Lviv, 1988. (in Russian)

[12] Levin B.Ya. Distribution of zeros of entire functions. In: Transl. Math. Monogr., 5, Amer. Math. Soc., Providence, R.I., 1980.

[13] Vasylkiv Ya.V. Asymptotic behavior of logarithmic derivatives and logarithms of meromorphic functions of completely regular growth in the metric of $L^p[0, 2π]$. I. Mat. Stud. 1999, 12 (1), 37-58. (in Ukrainian)

[14] Vynnyts'kyi B.V., Khats' R.V. On asymptotic behavior of entire functions of noninteger order. Mat. Stud. 2004, 21 (2), 140-150. (in Ukrainian)

[15] Vynnyts'kyi B.V., Khats' R.V. On growth regularity of entire function of noninteger order with zeros on a finite system of rays. Mat. Stud. 2005, 24 (1), 31-38. (in Ukrainian)