Composition of entire functions and bounded L-index in direction

A. I. Bandura

doi:10.15330/ms.47.2.179-184

In the present paper we give an answer to the following question: Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ be an entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What are a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction $\mathbf{b}$?

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

2. 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.

3. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109. doi:10.15330/ms.43.1.103–109

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

5. 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)

6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index in direction, Ukrain. Mat. J., 69 (2017), №3, 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. doi:10.17512/jamcm.2017.2.02

8. A. Bandura, O. Skaskiv, Analytic function in the unit ball, Beau Bassin: LAP Lambert Academic Publishing, 2017, 100 p. https://arxiv.org/abs/1705.09568

9. 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. Mathematica Bohemica (accepted for publication) https://arxiv.org/abs/1609.04190

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

11. V.O. Kushnir, Analogue of Hayman theorem for analytic functions of bounded l-index, Visn. Lviv Un-ty, Ser. Mekh.-Math., 53 (1999), 48–51. (in Ukrainian)

12. 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)

13. 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)

14. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Mat. Zametki, 39 (1986), №1, 3–13. Engl. transl.: Math. Notes, 39 (1986), №1, 3–8. doi:10.1007/BF01647624

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

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

17. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded lindex, Sib. Mat. Zh., 33 (1992), №2, 142–150. Engl. transl.: Sib. Math. J., 33 (1992), №2, 304–312. doi:10.1007/BF00971102

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

	Composition of entire functions and bounded L-index in direction
Author	A. I. Bandura andriykopanytsia@gmail.com Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
Abstract	In the present paper we give an answer to the following question: Let $f\colon \mathbb{C}\to \mathbb{C}$ be an entire function of bounded $l$-index, $\Phi\colon \mathbb{C}^n\to \mathbb{C}$ be an entire function, $n\geq2,$ $l\colon \mathbb{C}\to \mathbb{R}_+$ be a continuous function. What are a positive continuous function $L\colon \mathbb{C}^n\to \mathbb{R}_+$ and a direction $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ such that the composite function $f(\Phi(z))$ has bounded $L$-index in the direction $\mathbf{b}$?
Keywords	entire function; bounded L-index in direction; composite function; bounded l-index
DOI	doi:10.15330/ms.47.2.179-184
Reference	1. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian) 2. 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. 3. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109. doi:10.15330/ms.43.1.103–109 4. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. 5. 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) 6. A.I. Bandura, O.B. Skaskiv, Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index in direction, Ukrain. Mat. J., 69 (2017), №3, 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. doi:10.17512/jamcm.2017.2.02 8. A. Bandura, O. Skaskiv, Analytic function in the unit ball, Beau Bassin: LAP Lambert Academic Publishing, 2017, 100 p. https://arxiv.org/abs/1705.09568 9. 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. Mathematica Bohemica (accepted for publication) https://arxiv.org/abs/1609.04190 10. W.K. Hayman, Differential inequalities and local valency, Pacific J. Math., 44 (1973), №1, 117–137. 11. V.O. Kushnir, Analogue of Hayman theorem for analytic functions of bounded l-index, Visn. Lviv Un-ty, Ser. Mekh.-Math., 53 (1999), 48–51. (in Ukrainian) 12. 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) 13. 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) 14. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Mat. Zametki, 39 (1986), №1, 3–13. Engl. transl.: Math. Notes, 39 (1986), №1, 3–8. doi:10.1007/BF01647624 15. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307. 16. M.N. Sheremeta, Entire functions and Dirichlet series of bounded l-index, Russian Math. (Iz. VUZ), 36 (1992), №9, 76–82. 17. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded lindex, Sib. Mat. Zh., 33 (1992), №2, 142–150. Engl. transl.: Sib. Math. J., 33 (1992), №2, 304–312. doi:10.1007/BF00971102 18. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p.
Pages	179-184
Volume	47
Issue	2
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html