Extreme problems in the space of meromorphic functions of finite order in the half plane. II

K.G. Malyutin; A.A. Revenko

doi:10.30970/ms.54.2.154-161

K.G. Malyutin Kursk State University, Department of Mathematical Analysis, Kursk
A.A. Revenko Kursk State University, Department of Mathematical Analysis, Kursk

DOI: https://doi.org/10.30970/ms.54.2.154-161

Анотація

The extremal problems in the space of meromorphic functions of order $\rho>0$ 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 $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.

Посилання

K.G. Malyutin, A.A. Revenko, Extreme problems in the space of meromorphic functions of finite order in the half-plane, Mat. Stud., 52 (2019), 144–155. doi:10.30970/ms.52.2.144-155

L.A. Rubel, B.A. Taylor, A Fourier series method for meromorphic and entire function, Bull. Soc. Math. France, 96 (1968), 53–96.

A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. I, Mat. Sb., 106 (1978), No148, 386–408; English transl. in Math. USSR-Sb., 35 (1979).

A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. II, Mat. Sb., 113 (1980), No155, 118–132; English transl. in Math. USSR-Sb. 41 (1982).

A.A. Kondratyuk, The Fourier series method for entire and meromorphic functions of completely regular growth. III, Mat. Sb., 120 (1983), No162, 331–343; English transl. in Math. USSR-Sb., 48 (1984).

K.G. Malyutin, Fourier series and δ-subharmonic functions of finite γ-type in a half-plane, Sbornik: Mathematics, 192 (2001), No6, 843–861. doi:10.1070/SM2001v192n06ABEH000572

J.B. Miles , D.P. Shea, An extremal problem in value distribution theory, Quart. J. Math. Oxford, 24 (1973), 377–383.

A.A. Kondratyuk, Fourier Series and Meromorphic Functions, Vyshcha shkola, Lviv, 1988. (in Russian)

M.A. Fedorov, A.F. Grishin, Some questions of the Nevanlinna theory for the complex half-plane, Mathematical Physics, Analysis and Geometry, 1 (1998), No3, 223–271.

K.G. Malyutin, N. Sadik, Representation of subharmonic functions in a half-plane, Sbornik: Mathematics, 198 (2007), No12, 1747–1761. doi:10.1070/SM2007v198n12ABEH00390

G. Pólya, Untersuchungen über Lücen and Singularitätin von Potenzreihen, Math. Zeitschrift, 29 (1929), 549–640.

J. Bak, D.J. Newman, Complex Analysis, Springer, 2010.