Extreme problems in the space of meromorphic functions of finite order in the half-plane |
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| Author |
malyutinkg@gmail.com1, revenko253@mail.ru2
Kursk State University,
Department of Mathematical Analysis,
Kursk, Russia
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| Abstract |
Extreme problems in the space of meromorphic functions
of finite order in the half plane% A title of the paper (in English).
The extremal problems in the space of meromorphic functions of order $\rho>1$ in upper half-plane are studed.
The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>1$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, accurate estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained. The inequality related to the some analogue of the Nevanlinna problem for half-plane is proved.
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| Keywords |
extremal problem; meromorphic function of finite order; complete measure; Polya lemma;
Carleman formula; Nevanlinna characteristic; Parseval equality
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| DOI |
doi:10.30970/ms.52.2.144-155
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| Reference |
1. L.A. Rubel, B.A. Taylor, A Fourier series method for meromorphic and entire function, Bull. Soc. Math.
France, 96 (1968), 53.96.
2. A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. I, Mat. Sb. 106 (1978), ¹148, 386.408; English transl. in Math. USSR-Sb., 35 (1979). 3. A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. II, Mat. Sb. 113 (1980), ¹155, 118.132; English transl. in Math. USSR-Sb., 41 (1982). 4. A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. III, Mat. Sb. 120 (1983), ¹162, 331.343; English transl. in Math. USSR-Sb., 48 (1984). 5. K.G. Malyutin, Fourier series and $\delta$-subharmonic functions of finite -type in a half-plane, Sbornik: Mathematics, 192 (2001), ¹6, 843-861. doi : 10.1070/SM2001v192n06ABEH000572 6. J.B. Miles, D.P. Shea, An extremal problem in value distribution theory, Quart. J. Math. Oxford, 24 (1973), 377-383. 7. A.A. Kondratyuk, Fourier Series and Meromorphic Functions, Vyshcha shkola, Lviv. (in Russian) 8. M.A. Fedorov, A.F. Grishin, Some Questions of the Nevanlinna Theory for the Complex Half-Plane, Mathematical Physics, Analysis and Geometry, 1 (1998), ¹3, 223-271. 9. K.G. Malyutin, N. Sadik, Representation of subharmonic functions in a half-plane, Sbornik: Mathematics, 198 (2007), ¹12, 1747-1761. doi : 10.1070/SM2007v198n12ABEH00390 10. Polya G., Untersuchungen uber Nucen and Singularitatin von Potenzreihen, Math. Zeitschrift, 1929, 549-640. 11. J. Bak, D. J. Newman, Complex Analysis, Springer, 2010. |
| Pages |
144-155
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| Volume |
52
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| Issue |
2
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| Year |
2019
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| Journal |
Matematychni Studii
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| Full text of paper | |
| Table of content of issue |