Analytic in a polydisc functions of bounded
L-index in joint variables

A. I. Bandura, N. V. Petrechko, O. B. Skaskiv; O. B. Skaskiv; O. B. Skaskiv

doi:10.15330/ms.46.1.72-80

A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded L-index, Mat. Stud. 4 (1995), 53–58; Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26) is generalised for an analytic function in a polydisc. We proved criteria of boundedness of L-index in joint variables which describe local behaviour of partial derivative. Some improvements of known sufficient conditions of boundednees of L-index in joint variables are obtained.

1. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298–307.

2. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26.

3. M.T. Bordulyak, M.M. Sheremeta, Boundedness of the L-index of an entire function of several variables, Dopov. Akad. Nauk Ukr., (1993), №9, 10–13. (in Ukrainian)

4. M.T. Bordulyak, The space of entire in Cn functions of bounded L-index// Mat. Stud., 4 (1995), 53–58. (in Ukrainian)

5. S.N. Strochyk, M.M. Sheremeta, Analytic in the unit disc functions of bounded index, Dopov. Akad. Nauk Ukr., (1993), №1, 19–22. (Ukrainian)

6. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12 (1999), №1, 59–66.

7. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p.

8. Gopala J. Krishna, S.M. Shah, Functions of bounded indices in one and several complex variables, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, 223–235.

9. A. Bandura, O. Skaskiv, Analytic in the unit ball functions of bounded L-index in direciton, (submitted in Rocky Mountain Journal of Mathematics) https://arxiv.org/abs/1501.04166.

10. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian)

11. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. http://chyslo.com.ua

12. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), №2, 211–215. (in Ukrainian)

13. A.I. Bandura, O.B. Skaskiv, Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), №2, 177–182.

14. A.I. Bandura, On boundedness of the L-index in the direction for entire functions with plane zeros, Mat. Visn. Nauk. Tov. Im. Shevchenka, 6 (2009), 44–49. (in Ukrainian)

15. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109.

16. F. Nuray, R.F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci., 5 (2015), №2, 171–177.

17. F. Nuray, R.F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), №2, 225–233.

18. M. Salmassi, Functions of bounded indices in several variables, Indian J. Math., 31 (1989), №3, 249–257.

19. B.C. Chakraborty, R. Chanda, A class of entire functions of bounded index in several variables, J. Pure Math. 12 (1995), 16–21.

20. B.C. Chakraborty, Tapas Kumar Samanta, On entire functions of bounded index in several variables, J. Pure Math., 17 (2000), 53–71.

21. B.C. Chakraborty, Tapas Kumar Samanta, On entire functions of L-bounded index, J. Pure Math., 18 (2001), 53–64.

22. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J., 33 (1992), №2, 304–312.

	Analytic in a polydisc functions of bounded L-index in joint variables
Author	A. I. Bandura, N. V. Petrechko, O. B. Skaskiv andriykopanytsia@gmail.com, petrechko.n@gmail.com, olskask@gmail.com Ivano-Frankivsk National Technical University of Oil and Gas, Ivan Franko National University of Lviv
Abstract	A concept of boundedness of L-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded L-index, Mat. Stud. 4 (1995), 53–58; Bandura A. I., Bordulyak M. T., Skaskiv O. B. Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud. 45 (2016), 12–26) is generalised for an analytic function in a polydisc. We proved criteria of boundedness of L-index in joint variables which describe local behaviour of partial derivative. Some improvements of known sufficient conditions of boundednees of L-index in joint variables are obtained.
Keywords	analytic function; polydisc; bounded L-index in joint variables; maximum modulus; partial derivative
DOI	doi:10.15330/ms.46.1.72-80
Reference	1. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298–307. 2. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26. 3. M.T. Bordulyak, M.M. Sheremeta, Boundedness of the L-index of an entire function of several variables, Dopov. Akad. Nauk Ukr., (1993), №9, 10–13. (in Ukrainian) 4. M.T. Bordulyak, The space of entire in Cn functions of bounded L-index// Mat. Stud., 4 (1995), 53–58. (in Ukrainian) 5. S.N. Strochyk, M.M. Sheremeta, Analytic in the unit disc functions of bounded index, Dopov. Akad. Nauk Ukr., (1993), №1, 19–22. (Ukrainian) 6. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12 (1999), №1, 59–66. 7. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 8. Gopala J. Krishna, S.M. Shah, Functions of bounded indices in one and several complex variables, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, 223–235. 9. A. Bandura, O. Skaskiv, Analytic in the unit ball functions of bounded L-index in direciton, (submitted in Rocky Mountain Journal of Mathematics) https://arxiv.org/abs/1501.04166. 10. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian) 11. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. http://chyslo.com.ua 12. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), №2, 211–215. (in Ukrainian) 13. A.I. Bandura, O.B. Skaskiv, Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), №2, 177–182. 14. A.I. Bandura, On boundedness of the L-index in the direction for entire functions with plane zeros, Mat. Visn. Nauk. Tov. Im. Shevchenka, 6 (2009), 44–49. (in Ukrainian) 15. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109. 16. F. Nuray, R.F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci., 5 (2015), №2, 171–177. 17. F. Nuray, R.F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), №2, 225–233. 18. M. Salmassi, Functions of bounded indices in several variables, Indian J. Math., 31 (1989), №3, 249–257. 19. B.C. Chakraborty, R. Chanda, A class of entire functions of bounded index in several variables, J. Pure Math. 12 (1995), 16–21. 20. B.C. Chakraborty, Tapas Kumar Samanta, On entire functions of bounded index in several variables, J. Pure Math., 17 (2000), 53–71. 21. B.C. Chakraborty, Tapas Kumar Samanta, On entire functions of L-bounded index, J. Pure Math., 18 (2001), 53–64. 22. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J., 33 (1992), №2, 304–312.
Pages	72-80
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Analytic in a polydisc functions of bounded L-index in joint variables