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Home » Issues » Archive of Issues » 2020 - 3 issue » pp. 396-409

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Russia Kursk
Year
2020
Volume
30
Issue
3
Pages
396-409
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Section Mathematics
Title The meromorphic functions of completely regular growth on the upper half-plane
Author(-s) Malyutin K.G.a, Kabanko M.V.a
Affiliations Kursk State Universitya
Abstract A strictly positive continuous unbounded increasing function $\gamma(r)$ on the half-axis $[0,+\infty)$ is called growth function. Let the growth function $\gamma(r)$ satisfies the condition $\gamma(2r)\leq M\gamma(r)$ for some $M>0$ and for all $r>0$. In the paper, the class $JM(\gamma(r))^o$ of meromorphic functions of completely regular growth on the upper half-plane with respect to the growth function $\gamma$ is considered. The criterion for the meromorphic function $f$ to belong to the space $JM(\gamma(r))^o$ is obtained. The definition of the indicator of function from the space $JM(\gamma(r))^o$ is introduced. It is proved that the indicator belongs to the space $\mathbf{L}^p[0,\pi]$ for all $p>1$.
Keywords just meromorphic function, complete measure, function of growth, function of completely regular growth, Fourier coefficients, conjugate series, indicator
UDC 517.53
MSC 30D35, 30D30, 42A16, 30D15
DOI 10.35634/vm200304
Received 12 April 2020
Language English
Citation Malyutin K.G., Kabanko M.V. The meromorphic functions of completely regular growth on the upper half-plane, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 3, pp. 396-409.
References
  1. Bari N.K. Trigonometricheskie ryady (Trigonometric series), Moscow: GIFFL, 1961.
  2. Edwards R.E. Fourier series. A modern introduction. Vol. 2, New York, Heidelberg, Berlin: Springer-Verlag, 1982.
  3. Fedorov M.A., Grishin A.F. Some questions of the Nevanlinna theory for the complex half-plane, Interface Science, 1998, vol. 6, no. 3, pp. 223-271. https://doi.org/10.1023/A:1008672404744
  4. Govorov N.V. On the indicator of functions of nonintegral order that are analytic and of completely regular growth in a halfplane, Sov. Math., Dokl., 1965, vol. 6. pp. 697-701. https://zbmath.org/?q=an:0156.29604
  5. Govorov N.V. On the indicator of functions of integral order, analytic and of completely regular growth in a halfplane, Sov. Math., Dokl., 1967, vol. 8, pp. 159-163. https://zbmath.org/?q=an:0167.35701
  6. Grishin A.F. On regularity of the growth of subharmonic functions, Teoriya Funktsii, Funktsional'nyi Analiz i ikh Prilozheniya: Respublikanskii Nauchnyi Sbornik, issue 6, Kharkiv: Kharkiv University, 1968, pp. 3-29 (in Russian). http://dspace.univer.kharkov.ua/handle/123456789/2004
  7. Grishin A.F. On regularity of the growth of subharmonic functions. III, Teoriya Funktsii, Funktsional'nyi Analiz i ikh Prilozheniya: Respublikanskii Nauchnyi Sbornik, issue 7, Kharkiv: Kharkiv University, 1968, pp. 59-84 (in Russian). http://dspace.univer.kharkov.ua/handle/123456789/1915
  8. Grishin A.F. On regularity of the growth of subharmonic functions. IV, Teoriya Funktsii, Funktsional'nyi Analiz i ikh Prilozheniya: Respublikanskii Nauchnyi Sbornik, issue 8, Kharkiv: Kharkiv University, 1969, pp. 126-135 (in Russian). http://dspace.univer.kharkov.ua/handle/123456789/2034
  9. Grishin A.F. Continuity and asymptotic continuity of subharmonic functions. I, Matematicheskaya Fizika, Analiz, Geometriya, 1994, vol. 1, no. 2, pp. 193-215 (in Russian). http://jmag.ilt.kharkov.ua/jmag/pdf/1/m01-0193r.pdf
  10. Kondratjuk A.A. The Fourier series method for entire and meromorphic functions of completely regular growth, Mathematics of the USSR - Sbornik, 1979, vol. 35, no. 1, pp. 63-84. https://doi.org/10.1070/SM1979v035n01ABEH001452
  11. Kondratjyuk A.A. The Fourier series method for entire and meromorphic functions of completely regular growth. II, Mathematics of the USSR - Sbornik, 1982, vol. 41, no. 1, pp. 101-113. https://doi.org/10.1070/SM1982v041n01ABEH002223
  12. Kondratyuk A.A. The Fourier series method for entire and meromorphic functions of completely regular growth. III, Mathematics of the USSR - Sbornik, 1984, vol. 48, no. 2, pp. 327-338. https://doi.org/10.1070/SM1984v048n02ABEH002677
  13. Lévine B. Sur la croissance d'une fonction entière suivant un rayon et la distribution de ses zéros suivant leurs arguments, Rec. Math. [Mat. Sbornik] N.S., 1937, vol. 2 (44), no. 6, pp. 1097-1142 (in Russian). http://mi.mathnet.ru/eng/msb5644
  14. Levin B. Distribution of zeros of entire functions, Providence, RI: Amer. Math. Soc., 1964. https://doi.org/10.1090/mmono/005
  15. Malyutin K.G. Fourier series and $\delta$-subharmonic functions of finite $\gamma$-type in a half-plane, Sbornik: Mathematics, 2001, vol. 192, no. 6, pp. 843-861. https://doi.org/10.1070/SM2001v192n06ABEH000572
  16. Malyutin K.G., Sadik N. Delta-subharmonic functions of completely regular growth in the half-plane, Doklady Mathematics, 2001, vol. 64, no. 2, pp. 194-196. https://www.elibrary.ru/item.asp?id=27774718
  17. Malyutin K.G., Sadik N. Representation of subharmonic functions in a half-plane, Sbornik: Mathematics, 2007, vol. 198, no. 12, pp. 1747-1761. https://doi.org/10.1070/SM2007v198n12ABEH003904
  18. Pfluger A. Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Functionen. I, Commentarii Mathematici Helvetici, 1938, vol. 11, issue 1, pp. 180-214. https://doi.org/10.1007/BF01199698
  19. Pfluger A. Die Wertverteilung und das Verhalten von Betrag und Argument einer speziellen Klasse analytischer Functionen. II, Commentarii Mathematici Helvetici, 1939, vol. 12, issue 1, pp. 25-65. https://doi.org/10.1007/BF01620637
  20. Pólya G. Bemerkugen über unendlichen Folgen und ganze Functionen, Mathematische Annalen, 1923, vol. 88, issue 3, pp. 169-183. https://doi.org/10.1007/BF01579177
  21. Popov A.Yu., Solodov A.P. Estimates with sharp constants of the sums of sine series with monotone coefficients of certain classes in terms of the Salem majorant, Mathematical Notes, 2018, vol. 104, no. 5, pp. 702-711. https://doi.org/10.1134/S0001434618110111
  22. Rubel L.A., Taylor B.A. A Fourier series method for meromorphic and entire functions, Bulletin de la Société Mathématique de France, 1968, vol. 96, pp. 53-96. https://doi.org/10.24033/bsmf.1660
  23. Solodov A.P. Exact constants in Telyakovskii's two-sided estimate of the sum of a sine series with convex sequence of coefficients, Mathematical Notes, 2020, vol. 107, no. 6, pp. 988-1001. https://doi.org/10.1134/S0001434620050314
  24. Telyakovskii S.A. On the rate of convergence in $L$ of Fourier sine series with monotone coefficients, Mathematical Notes, 2016, vol. 100, no. 3, pp. 472-476. https://doi.org/10.1134/S0001434616090145
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