Composition, product and sum of analytic functions of bounded L-index in
direction in the unit ball

A. I. Bandura

doi:10.15330/ms.50.2.115-134

In this paper, we investigate a composition of entire function of one variable and analytic function in the unit ball. There are obtained conditions which provide equivalence of bounded\-ness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L\colon \mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l\colon \mathbb{C}\to \mathbb{R}_+.$ We present sufficient conditions for boundedness of $L$-index in the direction for sum and for product of functions analytic in the unit ball. The class of analytic functions in the unit ball having bounded $L$-index in direction is very wide because it contains all analytic functions with bounded multiplicities of zeros on every complex line $\{z^0+t\mathbf{b}\colon t\in\mathbb{C}\}$. It is a statement of proved existence theorem. In the one-dimensional case these results are new for functions analytic in the unit disc.

1. A.I. Bandura, Sum of entire functions of bounded L-index in direction, Mat. Stud., 45 (2016), №2, 149-158. doi: 10.15330/ms.45.2.149-158

2. A.I. Bandura, O.B. Skaskiv, Analytic functions in the unit ball and sufficient sets of boundedness of L-index in direction, Bukovyn. Mat. Zh., 6 (2018), №1-2, 13-20.

3. A.I. Bandura, N.V. Petrechko, Sum of entire functions of bounded index in joint variables, Electr. J. Math. Anal. Appl., 6 (2018), №2, 60-67.

4. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci., 227(1) (2017), 1-12. doi: 10.1007/s10958-017-3570-6

5. A.I. Bandura, O.B. Skaskiv, Analytic in an unit ball functions of bounded L-index in direction, 2015. arXiv: 1501.04166v2

6. A. Bandura, O. Skaskiv, Functions analytic in the unit ball having bounded L-index in a direction, to appear in Rocky Mountain J. Math. https://projecteuclid.org/euclid.rmjm/1542942029

7. A.I. Bandura, Analytic functions in the unit ball of bounded value L-distribution in a direction, Mat. Stud., 49 (2018), №1, 75-79. doi:10.15330/ms.49.1.75-79

8. A.I. Bandura, Product of two entire functions of bounded L-index in direction is a function with the same class, Bukovyn. Mat. Zh., 4 (2016), №1-2, 8-12.

9. A. Bandura, O. Skaskiv, Sufficient conditions of boundedness of L-index and analog of Haymans Theorem for analytic functions in a ball, Stud. Univ. Babes-Bolyai Math., 63 (2018), №4, 483-501. doi:10.24193/subbmath.2018.4.06

10. A. Bandura, O. Skaskiv, Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes, Demonstratio Mathematica, 52 (2019), №1, 82-87. doi:10.1515/dema-2019-0008

11. A.I. Bandura, O.B. Skaskiv, Analytic functions in the unit ball of bounded L-index: asymptotic and local properties, Mat. Stud., 48 (2017), №1, 37-73. doi: 10.15330/ms.48.1.37-73

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

13. A.I. Bandura, O.B. Skaskiv, Boundedness of L-index in direction of functions of the form f(.z;m.) and existence theorems, Mat. Stud., 41 (2014), №1, 45-52.

14. A.I. Bandura, O.B. Skaskiv, Boundedness of the L-index in a direction of entire solutions of second order partial differential equation, Acta Comment. Univ. Tartu. Math., 22 (2018), №2, 223-234. doi: 10.12697/ACUTM.2018.22.18

15. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p.

16. A. Bandura, Composition of entire functions and bounded L-index in direction, Mat. Stud., 47 (2017), №2, 179-184. doi:10.15330/ms.47.2.179.184

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

18. V.O. Kushnir, On analytic in a disc functions of bounded l-index, Visn. Lviv Un-ty, Ser. Mekh.-Math., 58 (2000), 21-24. (in Ukrainian)

19. V.O. Kushnir, Analytic function of bounded l-index: diss. ... Cand. Phys. and Math. Sciences, Ivan Franko National University of Lviv, Lviv, 2002, 132 p. (in Ukrainian)

20. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3-8. doi:10.1007/BF01647624

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

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

23. 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. doi:10.1007/BF00971102

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

25. 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)

26. W.J. Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc., 22 (1969), 319–323.

27. F. Nuray, R.F. Patterson, Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), №1, 67–74, doi: 10.15330/ms.49.1.67-74

	Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball
Author	A. I. Bandura andriykopanytsia@gmail.com Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
Abstract	In this paper, we investigate a composition of entire function of one variable and analytic function in the unit ball. There are obtained conditions which provide equivalence of bounded\-ness of $L$-index in a direction for such a composition and boundedness of $l$-index of initial function of one variable, where the continuous function $L\colon \mathbb{B}^n\to \mathbb{R}_+$ is constructed by the continuous function $l\colon \mathbb{C}\to \mathbb{R}_+.$ We present sufficient conditions for boundedness of $L$-index in the direction for sum and for product of functions analytic in the unit ball. The class of analytic functions in the unit ball having bounded $L$-index in direction is very wide because it contains all analytic functions with bounded multiplicities of zeros on every complex line $\{z^0+t\mathbf{b}\colon t\in\mathbb{C}\}$. It is a statement of proved existence theorem. In the one-dimensional case these results are new for functions analytic in the unit disc.
Keywords	bounded index; bounded L-index in direction; analytic function; unit ball; composite function; bounded l-index; sum; existence theorem
DOI	doi:10.15330/ms.50.2.115-134
Reference	1. A.I. Bandura, Sum of entire functions of bounded L-index in direction, Mat. Stud., 45 (2016), №2, 149-158. doi: 10.15330/ms.45.2.149-158 2. A.I. Bandura, O.B. Skaskiv, Analytic functions in the unit ball and sufficient sets of boundedness of L-index in direction, Bukovyn. Mat. Zh., 6 (2018), №1-2, 13-20. 3. A.I. Bandura, N.V. Petrechko, Sum of entire functions of bounded index in joint variables, Electr. J. Math. Anal. Appl., 6 (2018), №2, 60-67. 4. A. Bandura, O. Skaskiv, Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci., 227(1) (2017), 1-12. doi: 10.1007/s10958-017-3570-6 5. A.I. Bandura, O.B. Skaskiv, Analytic in an unit ball functions of bounded L-index in direction, 2015. arXiv: 1501.04166v2 6. A. Bandura, O. Skaskiv, Functions analytic in the unit ball having bounded L-index in a direction, to appear in Rocky Mountain J. Math. https://projecteuclid.org/euclid.rmjm/1542942029 7. A.I. Bandura, Analytic functions in the unit ball of bounded value L-distribution in a direction, Mat. Stud., 49 (2018), №1, 75-79. doi:10.15330/ms.49.1.75-79 8. A.I. Bandura, Product of two entire functions of bounded L-index in direction is a function with the same class, Bukovyn. Mat. Zh., 4 (2016), №1-2, 8-12. 9. A. Bandura, O. Skaskiv, Sufficient conditions of boundedness of L-index and analog of Haymans Theorem for analytic functions in a ball, Stud. Univ. Babes-Bolyai Math., 63 (2018), №4, 483-501. doi:10.24193/subbmath.2018.4.06 10. A. Bandura, O. Skaskiv, Analytic functions in the unit ball of bounded L-index in joint variables and of bounded L-index in direction: a connection between these classes, Demonstratio Mathematica, 52 (2019), №1, 82-87. doi:10.1515/dema-2019-0008 11. A.I. Bandura, O.B. Skaskiv, Analytic functions in the unit ball of bounded L-index: asymptotic and local properties, Mat. Stud., 48 (2017), №1, 37-73. doi: 10.15330/ms.48.1.37-73 12. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30-52. (in Ukrainian) 13. A.I. Bandura, O.B. Skaskiv, Boundedness of L-index in direction of functions of the form f(.z;m.) and existence theorems, Mat. Stud., 41 (2014), №1, 45-52. 14. A.I. Bandura, O.B. Skaskiv, Boundedness of the L-index in a direction of entire solutions of second order partial differential equation, Acta Comment. Univ. Tartu. Math., 22 (2018), №2, 223-234. doi: 10.12697/ACUTM.2018.22.18 15. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. 16. A. Bandura, Composition of entire functions and bounded L-index in direction, Mat. Stud., 47 (2017), №2, 179-184. doi:10.15330/ms.47.2.179.184 17. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117-137. 18. V.O. Kushnir, On analytic in a disc functions of bounded l-index, Visn. Lviv Un-ty, Ser. Mekh.-Math., 58 (2000), 21-24. (in Ukrainian) 19. V.O. Kushnir, Analytic function of bounded l-index: diss. ... Cand. Phys. and Math. Sciences, Ivan Franko National University of Lviv, Lviv, 2002, 132 p. (in Ukrainian) 20. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3-8. doi:10.1007/BF01647624 21. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298-307. 22. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76-82. 23. 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. doi:10.1007/BF00971102 24. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 25. 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) 26. W.J. Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc., 22 (1969), 319–323. 27. F. Nuray, R.F. Patterson, Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), №1, 67–74, doi: 10.15330/ms.49.1.67-74
Pages	115-134
Volume	50
Issue	2
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Composition, product and sum of analytic functions of bounded L-index in direction in the unit ball