Boundedness of l-index in direction of functions of the form
$f(\langle z,m\rangle)$ and existence theorems

A. I. Bandura; O. B. Skaskiv

We obtain a criterion of boundedness of $L$-index in direction for functions $f(\langle z,m\rangle)$. Using this criterion we find sufficient conditions of boundedness L-index in direction for some class of entire functions with ''plane'' zeros. Moreover, we prove some existence theorems of an entire function $f(\langle z,m\rangle)$ of bounded $L$-index in direction for a given $L$ and of a positive continuous function $L$ for a given entire function $F(z)$ such that $F$ is of bounded $L$-index in direction.

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

2. Bandura A.I., Skaskiv O.B. Entire function of bounded and unbounded L-index in direction// Mat. Stud. – 2007. – V.27, №2. – P. 211–215. (in Ukrainian)

3. Bandura A.I., Skaskiv O.B. Sufficient sets for boundedness L-index in direction for entire function// Mat. Stud. – 2008. – V.30, №2. – P. 177–182.

4. Bandura A.I. On boundedness of the L-index in direction for entire functions with plane zeros // Math. Bull. Shevchenko Sci. Soc. – 2008. – V.6. – P. 44–49. (in Ukrainian)

5. Papush D.E. On the growth of entire functions with “planar” zeros// Theory of functions, functional analysis and its application. – 1987. – V.48. – P. 117–125. (in Russian) English translated in Journal of Soviet Math. – 1990. – V.49, №2. – P. 930–935.

6. Bandura A.I. A modified criterion of boundedness of L-index in direction// Mat. Stud. – 2013. – V.39, №1. – P. 99–102.

7. Sheremeta M.M. On entire function and Dirichlet series of bounded l-index// Russ. Math. – V.36, №9. – 1992. – P. 76–82.

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

9. Goldberg A.A., Sheremeta M.M. On the boundedness of l-index canonical products// Ukr. Math. Bull. – 2005. – V.2, №1. – P. 52–54. (in Ukrainian)

10. Chyzhykov I.E., Sheremeta M.M. Boundedness of l-index for entire functions of zero genus// Mat. Stud. – 2001. – V.16, №2. – P. 124–130.

11. Sheremeta M.M. Generalization of the Fricke theorem on entire functions of bounded index// Ukr. Math. Journ. – 1996. – V.48, №3. – P. 460–466.

12. Bordulyak M.T., Sheremeta M.N. On the existence of entire functions of bounded l-index and l-regular growth// Ukr. Math. Journ. – 1996. – V.48, №9. – P. 1322–1340.

13. Goldberg A.A., Sheremeta M.M. Existence of an entire transcendental function of bounded l-index// Math. Notes. – 1995. – V.57, №1–2. – P. 88–90.

14. Bordulyak M.T. A proof of Sheremeta conjecture concerning entire function of bounded l-index// Mat. Stud. – 1999. – V.11, №2. – P. 108–110.

15. Sheremeta M.M. Remark to existence theorem for entire function of bounded l-index// Mat. Stud. – 2000. – V.13, №1. – P. 97–99.

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

	Boundedness of l-index in direction of functions of the form $f(\langle z,m\rangle)$ and existence theorems
Author	A. I. Bandura, O. B. Skaskiv andriykopanytsia@gmail.com, skask@km.ru Ivano-Frankivsk National Technical University of Oil and Gas, Ivan Franko National University of Lviv
Abstract	We obtain a criterion of boundedness of $L$-index in direction for functions $f(\langle z,m\rangle)$. Using this criterion we find sufficient conditions of boundedness L-index in direction for some class of entire functions with ''plane'' zeros. Moreover, we prove some existence theorems of an entire function $f(\langle z,m\rangle)$ of bounded $L$-index in direction for a given $L$ and of a positive continuous function $L$ for a given entire function $F(z)$ such that $F$ is of bounded $L$-index in direction.
Keywords	entire function of several variables; bounded L-index in direction; existence theorem; entire function with “plane” zeros
Reference	1. Bandura A.I., Skaskiv O.B. Entire function of bounded L-index in direction// Mat. Stud. – 2007. – V.27, №1. – P. 30–52. (in Ukrainian) 2. Bandura A.I., Skaskiv O.B. Entire function of bounded and unbounded L-index in direction// Mat. Stud. – 2007. – V.27, №2. – P. 211–215. (in Ukrainian) 3. Bandura A.I., Skaskiv O.B. Sufficient sets for boundedness L-index in direction for entire function// Mat. Stud. – 2008. – V.30, №2. – P. 177–182. 4. Bandura A.I. On boundedness of the L-index in direction for entire functions with plane zeros // Math. Bull. Shevchenko Sci. Soc. – 2008. – V.6. – P. 44–49. (in Ukrainian) 5. Papush D.E. On the growth of entire functions with “planar” zeros// Theory of functions, functional analysis and its application. – 1987. – V.48. – P. 117–125. (in Russian) English translated in Journal of Soviet Math. – 1990. – V.49, №2. – P. 930–935. 6. Bandura A.I. A modified criterion of boundedness of L-index in direction// Mat. Stud. – 2013. – V.39, №1. – P. 99–102. 7. Sheremeta M.M. On entire function and Dirichlet series of bounded l-index// Russ. Math. – V.36, №9. – 1992. – P. 76–82. 8. Sheremeta M.M. Analytic functions of bounded index. – Lviv: VNTL Publishers, 1999. – 141 p. 9. Goldberg A.A., Sheremeta M.M. On the boundedness of l-index canonical products// Ukr. Math. Bull. – 2005. – V.2, №1. – P. 52–54. (in Ukrainian) 10. Chyzhykov I.E., Sheremeta M.M. Boundedness of l-index for entire functions of zero genus// Mat. Stud. – 2001. – V.16, №2. – P. 124–130. 11. Sheremeta M.M. Generalization of the Fricke theorem on entire functions of bounded index// Ukr. Math. Journ. – 1996. – V.48, №3. – P. 460–466. 12. Bordulyak M.T., Sheremeta M.N. On the existence of entire functions of bounded l-index and l-regular growth// Ukr. Math. Journ. – 1996. – V.48, №9. – P. 1322–1340. 13. Goldberg A.A., Sheremeta M.M. Existence of an entire transcendental function of bounded l-index// Math. Notes. – 1995. – V.57, №1–2. – P. 88–90. 14. Bordulyak M.T. A proof of Sheremeta conjecture concerning entire function of bounded l-index// Mat. Stud. – 1999. – V.11, №2. – P. 108–110. 15. Sheremeta M.M. Remark to existence theorem for entire function of bounded l-index// Mat. Stud. – 2000. – V.13, №1. – P. 97–99. 16. Bordulyak M.T., Sheremeta M.M. Boundedness of the L-index of an entire function of several variables// Dopov. Akad. Nauk Ukr. – 1993. – №9. – P. 10–13. (in Ukrainian)
Pages	45-52
Volume	41
Issue	1
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Boundedness of l-index in direction of functions of the form $f(\langle z,m\rangle)$ and existence theorems