Growth of analytic functions in the unit disc and complete measure in the sense of Grishin

I.E.Chyzhykov

Let $\rho_T[f] $ and $\rho_M[f]$ be the orders of an analytic function $f$ in the unit disc defined by the Nevanlinna characteristic and the maximum modulus function, respectively. Given $0\le \sigma \le \rho\le\sigma +1$, $\rho\ge 1$, we describe the class $A_\sigma^\rho$ of analytic function in $\mathbb{D}$ such that $\rho_T[f]=\sigma$, $\rho_M[f]=\rho$ in terms of the so called complete measure (in the sense of Grishin) of an analytic function. The proofs are based on results of C.N.Linden and recent results of the author.

1. Нафталефич А.Г. Об интерполировании функций, мероморфных в единичном круге, Докл. АН СССР. LXXXVIII (1953), № 2, 205--208.

2. Tsuji M. Canonical product for a meromorphic function in a unit circle, J.Math.Soc.Japan 8 (1956), №1, 7--21.

3. Linden C.N. The representation of regular functions, J. London Math. Soc. 39 (1964), 19--30.

4. Linden C.N. On a conjecture of Valiron concerning sets of indirect Borel point, J.~London Math.Soc. 41 (1966), 304--312.

5. Linden C.N. The minimum modulus of functions regular and of finite order in the unit circle, Quart.J.Math.(2) 7 (1956), 196--216.

6. Linden C.N. Integral logarithmic means for regular functions, Pacific J. of Math. 138 (1989), №1, 119--127.

7. Linden C.N. The characterization of orders for regular functions, Math. Proc. Cambrodge Phil. Soc. 111 (1992), №2, 299--307.

8. Чижиков I.E. Про повний опис класу аналітичних в крузі функцій без нулів із заданими величинами порядків, Укр. мат. журн. 59 (2007), №7, 979--995.

9. Джрбашян М.М. Интегральные преобразования и представления функций в комплексной области, М.: Наука, 1966.

10. Zygmund A. Trigonometric series, V.1,2 Cambridge Univ. Press, 1959.

11. Hardy G.H., Littlewood J.E. Some properties of fractional integrals. II, Math. Zeitschrift 34 (1931/32), 403--439.

12. Привалов И.И. Граничные свойства однозначных аналитических функций, М. МГУ. -- 1941.

13. Шамоян Ф.А. Несколько замечаний к параметрическому представлению классов Неванлинны-Джрбашяна, Мат. заметки 52 (1992), №1, 128--140.

14. Chyzhykov I. Growth and representation of analytic and harmonic functions in the unit disk, Ukr.Math.Bull. 3 (2006), №1, 31--44.

15. Гришин А. Непрерывность и асимптотическая непрерывность субгармонических функций, Мат. физика, анализ, геометрия 1 (1994), №2, 193--215.

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

17. Chyzhykov I. On the argument of bounded analytic functions, preprint.

18. Collingwood E.F., Lohwater A.J. The theory of claster sets, Cambridge Univer. Press, 1966.

19. Salem R. On a theorem of Zygmund, Duke Math.J. 10 (1943), 23--31.

	Growth of analytic functions in the unit disc and complete measure in the sense of Grishin
Author	I.E.Chyzhykov ichyzh@lviv.farlep.net Faculty of Mechanics and Mathematics, Ivan Franko National University of Lviv,
Abstract	Let $\rho_T[f] $ and $\rho_M[f]$ be the orders of an analytic function $f$ in the unit disc defined by the Nevanlinna characteristic and the maximum modulus function, respectively. Given $0\le \sigma \le \rho\le\sigma +1$, $\rho\ge 1$, we describe the class $A_\sigma^\rho$ of analytic function in $\mathbb{D}$ such that $\rho_T[f]=\sigma$, $\rho_M[f]=\rho$ in terms of the so called complete measure (in the sense of Grishin) of an analytic function. The proofs are based on results of C.N.Linden and recent results of the author.
Keywords	growth, analytic function, unit disc, complete measure
DOI	doi:10.30970/ms.29.1.35-44
Reference	1. Нафталефич А.Г. Об интерполировании функций, мероморфных в единичном круге, Докл. АН СССР. LXXXVIII (1953), № 2, 205--208. 2. Tsuji M. Canonical product for a meromorphic function in a unit circle, J.Math.Soc.Japan 8 (1956), №1, 7--21. 3. Linden C.N. The representation of regular functions, J. London Math. Soc. 39 (1964), 19--30. 4. Linden C.N. On a conjecture of Valiron concerning sets of indirect Borel point, J.~London Math.Soc. 41 (1966), 304--312. 5. Linden C.N. The minimum modulus of functions regular and of finite order in the unit circle, Quart.J.Math.(2) 7 (1956), 196--216. 6. Linden C.N. Integral logarithmic means for regular functions, Pacific J. of Math. 138 (1989), №1, 119--127. 7. Linden C.N. The characterization of orders for regular functions, Math. Proc. Cambrodge Phil. Soc. 111 (1992), №2, 299--307. 8. Чижиков I.E. Про повний опис класу аналітичних в крузі функцій без нулів із заданими величинами порядків, Укр. мат. журн. 59 (2007), №7, 979--995. 9. Джрбашян М.М. Интегральные преобразования и представления функций в комплексной области, М.: Наука, 1966. 10. Zygmund A. Trigonometric series, V.1,2 Cambridge Univ. Press, 1959. 11. Hardy G.H., Littlewood J.E. Some properties of fractional integrals. II, Math. Zeitschrift 34 (1931/32), 403--439. 12. Привалов И.И. Граничные свойства однозначных аналитических функций, М. МГУ. -- 1941. 13. Шамоян Ф.А. Несколько замечаний к параметрическому представлению классов Неванлинны-Джрбашяна, Мат. заметки 52 (1992), №1, 128--140. 14. Chyzhykov I. Growth and representation of analytic and harmonic functions in the unit disk, Ukr.Math.Bull. 3 (2006), №1, 31--44. 15. Гришин А. Непрерывность и асимптотическая непрерывность субгармонических функций, Мат. физика, анализ, геометрия 1 (1994), №2, 193--215. 16. Fedorov M.A., Grishin A.F. Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Analysis and Geometry (Kluwer Acad. Publish.) 1 (1998), №3, 223--271. 17. Chyzhykov I. On the argument of bounded analytic functions, preprint. 18. Collingwood E.F., Lohwater A.J. The theory of claster sets, Cambridge Univer. Press, 1966. 19. Salem R. On a theorem of Zygmund, Duke Math.J. 10 (1943), 23--31.
Pages	35-44
Volume	29
Issue	1
Year	2008
Journal	Matematychni Studii
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