One criterion of $\gamma$-type finiteness of 
a function analytic in a half-plane

A. Ya. Khrystiyanyn

Let $\gamma$ be a growth function and let $f$ be an analytic in the closure of the upper half-plane function such that $|f(t)| \leq 1$ for real $t$. The classes of subharmonic functions of finite $\gamma$-type were introduced and studied by K.~G.~Malyutin. We prove a criterion of $\gamma$-type finiteness of $ \log |f|$ in the terms of the Fourier coefficients of $\arg f$.

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

2. Rubel L. A. Entire and meromorphic functions, New York: Springer, 1996, 200 pp.

3. Кондратюк А. А. Ряды Фурье и мероморфные функции, Львов, Выща школа, 1988, 195 с.

4. Malyutin K.\ G. Fourier series and $\delta$-subharmonic functions of finite ${\gamma}$-type in a half-plane, Math. Russ-Sb. 192 (2001), 846--861.

5. Гришин А. Ф. Непрерывность и асимптотическая непрерывность субгармонических функций, Мат. физика, анализ, геометрия, 1 (1994), 193–215.

6. Grishin A. F., Fedorov M. A. Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Analysis and Geometry, 1 (1998), № 3, 223–271.

7. Гольдберг А. А., Островский И. В. Распределение значений мероморфных функций, М.: Наука, 1970, 591 с.

8. Govorov N. V. Riemann's boundary problem with infinite index, Birkhäuser, Basel 1994.

	One criterion of $\gamma$-type finiteness of a function analytic in a half-plane
Author	A. Ya. Khrystiyanyn khrystiyanyn@ukr.net Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University,
Abstract	Let $\gamma$ be a growth function and let $f$ be an analytic in the closure of the upper half-plane function such that $\|f(t)\| \leq 1$ for real $t$. The classes of subharmonic functions of finite $\gamma$-type were introduced and studied by K.~G.~Malyutin. We prove a criterion of $\gamma$-type finiteness of $ \log \|f\|$ in the terms of the Fourier coefficients of $\arg f$.
Keywords	analytic function, half-plane, growth function
DOI	doi:10.30970/ms.21.2.151-169
Reference	1. Rubel L. A., Taylor B. A. Fourier series method for meromorphic and entire functions, Bull. Soc. Math. France, 96 (1968), 53–96. 2. Rubel L. A. Entire and meromorphic functions, New York: Springer, 1996, 200 pp. 3. Кондратюк А. А. Ряды Фурье и мероморфные функции, Львов, Выща школа, 1988, 195 с. 4. Malyutin K.\ G. Fourier series and $\delta$-subharmonic functions of finite ${\gamma}$-type in a half-plane, Math. Russ-Sb. 192 (2001), 846--861. 5. Гришин А. Ф. Непрерывность и асимптотическая непрерывность субгармонических функций, Мат. физика, анализ, геометрия, 1 (1994), 193–215. 6. Grishin A. F., Fedorov M. A. Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Analysis and Geometry, 1 (1998), № 3, 223–271. 7. Гольдберг А. А., Островский И. В. Распределение значений мероморфных функций, М.: Наука, 1970, 591 с. 8. Govorov N. V. Riemann's boundary problem with infinite index, Birkhäuser, Basel 1994.
Pages	151-169
Volume	21
Issue	2
Year	2004
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

One criterion of $\gamma$-type finiteness of a function analytic in a half-plane