Analytic functions in the unit ball of bounded L-index: asymptotic
and local properties

A. I. Bandura; O. B. Skaskiv

doi:10.15330/ms.48.1.37-73

We have generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic functions in the unit ball, where $\mathbf{L}\colon \mathbb{B}^n\to \mathbb{R}^n_+$ is a continuous vector-function, $\mathbb{B}^n$ is the unit ball in $\mathbb{C}^n.$ One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function $\mathbf{L}.$ Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives on sphere in $\mathbb{C}^n.$ The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of $\mathbf{L}$-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained.

1. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci. 227 (2017), №1, 1-12.

2. A. Bandura, O. Skaskiv, Sufficient conditions of boundedness of L-index and analog of HaymanЃfs Theorem for analytic functions in a ball, Studia Universitatis BabeЃCs-Bolyai Mathematica (in print), avialable at https://arxiv.org/abs/1705.09568.

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

4. A.I. Bandura, Sum of entire functions of bounded L-index in direction, Mat. Stud., 45 (2016), №2, 149-158.

5. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Publisher I.E.Chyzhykov, Lviv, 2016.

6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index along the direction, Ukrain. Mat. J., 69 (2017), №1, 500–508.

7. A. Bandura, O. Skaskiv, P. Filevych, Properties of entire solutions of some linear PDE’s, J. Appl. Math. Comput. Mech., 16 (2017), №2, 17–28.

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

9. 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.

10. A. Bandura, New criteria of boundedness of L-index in joint variables for entire functions, Math. Bull. Shevchenko Sci. Soc., 13 (2016), 58–67. (in Ukrainian)

11. A. Bandura, N. Petrechko, Properties of power series expansion of entire function of bounded L-index in joint variables, Visn. Lviv Un-ty, Ser. Mech. Math., 82 (2016), 27–33. (in Ukrainian)

12. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Analytic functions in a polydisc of bounded L-index in joint variables, Mat. Stud., 46 (2016), №1, 72–80.

13. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Maximum modulus of analytic in a bidisc functions of bounded L-index and analogue of Theorem of Hayman, Matematica Bohemica.

14. A.I. Bandura, N.V. Petrechko, Properties of power series of analytic in a bidisc functions of bounded L-index in joint variables, Carpathian Math. Publ., 9 (2017), №1, 6–12.

15. A.I. Bandura, Properties of positive continuous functions in $\mathbb{C}^n$, Carpathian Math. Publ., 7 (2015), №2, 137–147.

16. A.I. Bandura, Some improvements of criteria of L-index boundedness in direction, Mat. Stud., 47 (2017), №1, 27–32.

17. A. Bandura, O. Skaskiv, Entire functions of bounded L-Index: its zeros and behavior of partial logarithmic derivatives, J. Complex Analysis, Article ID 3253095, 10 p, 2017.

18. A.I. Bandura, O.B. Skaskiv, Iyer’s metric space, existence theorem and entire functions of bounded L-index in joint variables, Bukovyn. Mat. Zh., 5 (2017), №3-4, 8–14. (in Ukrainian)

19. M.T. Bordulyak, A proof of Sheremeta conjecture concerning entire function of bounded l-index, Mat. Stud., 11 (1999), №2, 108–110.

20. M.T. Bordulyak, On the growth of entire solutions of linear differential equations, Mat. Stud., 13 (2000), №2, 219–223.

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

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

23. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117–137.

24. G.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.

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

26. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307.

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

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

29. R. Patterson, F. Nuray, A characterization of holomorphic bivariate functions of bounded index, Mathematica Slovaca, 67 (2017), №3, 731–736.

30. L.I. Ronkin, Introduction to theory of entire functions of several variables, Nauka, Moscow, 1971, (in Russian); Engl. transl.: L.I. Ronkin, Introduction to theory of entire functions of several variables. AMS, Translations of mathematical monographs, V. 44, 1974.

31. W. Rudin, Function Theory in the unit ball on $\mathbb{C}^n$. – Reprint of the 1980 Edition, Springer, 2008.

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

33. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76–82.

34. 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.

35. M. Sheremeta, Analytic functions of bounded index, VNTL Publishers, Lviv, 1999.

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

37. K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, Springer, New York, 2005.

	Analytic functions in the unit ball of bounded L-index: asymptotic and local properties
Author	A. I. Bandura, O. B. Skaskiv andriykopanytsia@gmail.com, olskask@gmail.com Department of Advanced Mathematics, Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine; Department of Mechinics and Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	We have generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic functions in the unit ball, where $\mathbf{L}\colon \mathbb{B}^n\to \mathbb{R}^n_+$ is a continuous vector-function, $\mathbb{B}^n$ is the unit ball in $\mathbb{C}^n.$ One of propositions gives an estimate of the coefficients of power series expansions by a dominating homogeneous polynomial for analytic functions in the unit ball. Also we provide growth estimates of these functions. They describe the behavior of maximum modulus of analytic function on a skeleton in a polydisc by behavior of the function $\mathbf{L}.$ Most of our results are based on polydisc exhaustion of the unit ball. Nevertheless, we have generalized criteria of boundedness of $\mathbf{L}$-index in joint variables which describe local behavior of partial derivatives on sphere in $\mathbb{C}^n.$ The proposition uses a ball exhaustion. An analog of Hayman's theorem is applied to investigation of boundedness of $\mathbf{L}$-index in joint variables for analytic solutions in the unit ball of some linear higher-order systems of PDE's. There were found sufficient conditions providing the boundedness. Growth estimates of analytic solutions in the unit ball are also obtained.
Keywords	analytic function in a ball; bounded index in joint variables; maximum modulus; partial derivative; Cauchy’s integral formula; geometric exhaustion; growth estimates; linear higher-order systems of PDE
DOI	doi:10.15330/ms.48.1.37-73
Reference	1. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci. 227 (2017), №1, 1-12. 2. A. Bandura, O. Skaskiv, Sufficient conditions of boundedness of L-index and analog of HaymanЃfs Theorem for analytic functions in a ball, Studia Universitatis BabeЃCs-Bolyai Mathematica (in print), avialable at https://arxiv.org/abs/1705.09568. 3. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30-52. (in Ukrainian) 4. A.I. Bandura, Sum of entire functions of bounded L-index in direction, Mat. Stud., 45 (2016), №2, 149-158. 5. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Publisher I.E.Chyzhykov, Lviv, 2016. 6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index along the direction, Ukrain. Mat. J., 69 (2017), №1, 500–508. 7. A. Bandura, O. Skaskiv, P. Filevych, Properties of entire solutions of some linear PDE’s, J. Appl. Math. Comput. Mech., 16 (2017), №2, 17–28. 8. A. Bandura, O. Skaskiv, Analytic in the unit ball functions of bounded L-index in direction, (submitted in Rocky Mountain Journal of Mathematics), avialable at https://arxiv.org/abs/1501.04166. 9. 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. 10. A. Bandura, New criteria of boundedness of L-index in joint variables for entire functions, Math. Bull. Shevchenko Sci. Soc., 13 (2016), 58–67. (in Ukrainian) 11. A. Bandura, N. Petrechko, Properties of power series expansion of entire function of bounded L-index in joint variables, Visn. Lviv Un-ty, Ser. Mech. Math., 82 (2016), 27–33. (in Ukrainian) 12. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Analytic functions in a polydisc of bounded L-index in joint variables, Mat. Stud., 46 (2016), №1, 72–80. 13. A.I. Bandura, N.V. Petrechko, O.B. Skaskiv, Maximum modulus of analytic in a bidisc functions of bounded L-index and analogue of Theorem of Hayman, Matematica Bohemica. 14. A.I. Bandura, N.V. Petrechko, Properties of power series of analytic in a bidisc functions of bounded L-index in joint variables, Carpathian Math. Publ., 9 (2017), №1, 6–12. 15. A.I. Bandura, Properties of positive continuous functions in $\mathbb{C}^n$, Carpathian Math. Publ., 7 (2015), №2, 137–147. 16. A.I. Bandura, Some improvements of criteria of L-index boundedness in direction, Mat. Stud., 47 (2017), №1, 27–32. 17. A. Bandura, O. Skaskiv, Entire functions of bounded L-Index: its zeros and behavior of partial logarithmic derivatives, J. Complex Analysis, Article ID 3253095, 10 p, 2017. 18. A.I. Bandura, O.B. Skaskiv, Iyer’s metric space, existence theorem and entire functions of bounded L-index in joint variables, Bukovyn. Mat. Zh., 5 (2017), №3-4, 8–14. (in Ukrainian) 19. M.T. Bordulyak, A proof of Sheremeta conjecture concerning entire function of bounded l-index, Mat. Stud., 11 (1999), №2, 108–110. 20. M.T. Bordulyak, On the growth of entire solutions of linear differential equations, Mat. Stud., 13 (2000), №2, 219–223. 21. B.C. Chakraborty, R. Chanda, A class of entire functions of bounded index in several variables, J. Pure Math., 12 (1995), 16–21. 22. B.C. Chakraborty, T.K. Samanta, On entire functions of bounded index in several variables, J. Pure Math., 17 (2000), 53–71. 23. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117–137. 24. G.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. 25. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12 (1999), №1, 59–66. 26. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307. 27. F. Nuray, R.F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci., 5 (2015), №2, 171–177. 28. F. Nuray, R.F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), №2, 225–233. 29. R. Patterson, F. Nuray, A characterization of holomorphic bivariate functions of bounded index, Mathematica Slovaca, 67 (2017), №3, 731–736. 30. L.I. Ronkin, Introduction to theory of entire functions of several variables, Nauka, Moscow, 1971, (in Russian); Engl. transl.: L.I. Ronkin, Introduction to theory of entire functions of several variables. AMS, Translations of mathematical monographs, V. 44, 1974. 31. W. Rudin, Function Theory in the unit ball on $\mathbb{C}^n$. – Reprint of the 1980 Edition, Springer, 2008. 32. M. Salmassi, Functions of bounded indices in several variables, Indian J. Math., 31 (1989), №3, 249–257. 33. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76–82. 34. 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. 35. M. Sheremeta, Analytic functions of bounded index, VNTL Publishers, Lviv, 1999. 36. S.N. Strochyk, M.M. Sheremeta, Analytic in the unit disc functions of bounded index, Dopov. Akad. Nauk Ukr., (1993), №1, 19–22. (in Ukrainian) 37. K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, Springer, New York, 2005.
Pages	37-73
Volume	48
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Analytic functions in the unit ball of bounded L-index: asymptotic and local properties