Geodesic
The Fourier series method for entire and meromorphic functions of completely regular growth. II
Sbornik. Mathematics, Tome 41 (1982) no. 1, pp. 101-113 
A. A. Kondratyuk. The Fourier series method for entire and meromorphic functions of completely regular growth. II. Sbornik. Mathematics, Tome 41 (1982) no. 1, pp. 101-113. http://geodesic.mathdoc.fr/item/SM_1982_41_1_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Fourier series method is used to obtain an integral criterion for an entire function to be of completely regular growth. It is shown that when the pair $(Z,W)$ of sequences $Z$ of zeros and $W$ of poles of a meromorphic function $f$ has an angular density, the function belongs to the class $\Lambda^0$ of meromorphic functions of completely regular growth introduced in Part I of this paper, and the asymptotic properties of this function are studied. A function $f\in\Lambda^0$ for which $(Z,W)$ does not have an angular density is constructed; examples of $[\varkappa,\rho]$-trigonometrically convex functions are presented. Bibliography: 14 titles.

[1] A. A. Kondratyuk, “Metod ryadov Fure dlya tselykh i meromorfnykh funktsii vpolne regulyarnogo rosta”, Matem. sb., 106(148) (1978), 386–408 | MR | Zbl

[2] B. Ya. Levin, Raspredelenie kornei tselykh funktsii, Gostekhizdat, Moskva, 1956

[3] A. Zigmund, Trigonometricheskie ryady, t. 2, izd-vo “Mir”, Moskva, 1965 | MR

[4] L. A. Rubel, B. A. Taylor, “A Fourier series method for mesomorphic and entire functions”, Bull. Soc. Math. France, 96 (1968), 53–96 | Zbl

[5] A. A. Goldberg, “O defektakh tselykh funktsii vpolne regulyarnogo rosta”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, no. 14, Kharkov, 1971, 88–101

[6] S. Hellerstein, J. Williamson, “Entire functions with negative zeros and a problem of R. Nevanlinna”, J. analyse math., 22 (1969), 233–267 | DOI | MR | Zbl

[7] U. K. Kheiman, Meromorfnye funktsii, izd-vo “Mir”, Moskva, 1966 | MR

[8] A. A. Goldberg, I. V. Ostrovskii, Raspredelenie znachenii meromorfnykh funktsii, izd-vo “Nauka”, Moskva, 1970 | MR

[9] A. A. Kondratyuk, “Ekstremalnyi indikator dlya tselykh funktsii s polozhitelnymi nulyami”, Lit. matem. sb., 7:1 (1967), 79–117

[10] A. A. Goldberg, “Integral po poluadditivnoi mere i ego prilozhenie k teorii tselykh funktsii. IV”, Matem. sb., 66(108) (1965), 411–457

[11] A. A. Goldberg, “Integral po poluadditivnoi mere i ego prilozhenie k teorii tselykh funktsii. III”, Matem. sb., 65(107) (1964), 414–453 | MR

[12] Yu. P. Lapenko, O meromorfnykh funktsiyakh vpolne regulyarnogo rosta. I, Dep. VINITI, no. 2267–77, 1977

[13] G. Polya, “Untersuchungen iiber Zucken und Singularitaten von Potenzreihen”, Math. Z., 29 (1929), 549–640 | DOI | MR | Zbl

[14] A. A. Kondratyuk, “Tselye funktsii s polozhitelnymi nulyami, imeyuschimi konechnuyu maksimalnuyu plotnost”, Teoriya funktsii, funkts. analiz i ikh prilozheniya, no. 7, Kharkov, 1968, 37–52 | Zbl