# Some improvements of criteria of L-index boundedness in direction

Author
Ivano-Frankivsk National Technical University of Oil and Gas, Ivano-Frankivsk, Ukraine
Abstract
In this paper, we improve criteria of boundedness of $L$-index in direction for entire functions in $\mathbb{C}^n.$ They give an estimate of maximum modulus on circles of various radius, maximum modulus by minimum modulus on circle and describe the behavior of directional logarithmic derivative and the distribution of zeros. The obtained results are also new for entire functions of bounded index in $\mathbb{C}.$
Keywords
entire function; bounded L-index in direction; distribution of zeros; directional logarithmic derivative
DOI
doi:10.15330/ms.47.1.27-32
Reference
1. 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. Zh., 69 (2017), ¹3, 426–432. (in Ukrainian)

2. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), ¹1, 103–109.

3. G.H. Fricke, Entire functions of locally slow growth, J. Anal. Math., 28 (1975), ¹1, 101–122.

4. G.H. Fricke, Functions of bounded index and their logarithmic derivatives, Math. Ann., 206 (1973), 215–223.

5. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Mat. Zh., 33 (1992), ¹2, 142–150. Engl. transl.: Sib. Math. J., 33 (1992), ¹2, 304–312.

6. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), ¹1, 30–52. (in Ukrainian)

7. A.I. Bandura, A modified criterion of boundedness of L-index in direction, Mat. Stud., 39 (2013), ¹1, 99–102.

8. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), ¹1, 12–26.

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

10. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Mat. Zametki, 39 (1986), ¹1. – P. 3–13, (in Russian); Engl. transl.: Math. Notes, 39 (1986), ¹1, 3–8.

11. A.D. Kuzyk, M.N. Sheremeta, On entire functions, satisfying linear differential equations, Diff. equations, 26 (1990), ¹10, 1716–1722. (in Russian)

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

13. M.T. Bordulyak, M.M. Sheremeta, On the existence of entire functions of bounded l-index and l-regular growth, Ukrain. Mat. Zh., 48 (1996), ¹9, 1166–1182, (in Ukrainian); Engl. transl.: Ukrainian Math. J., 48 (1996), ¹9, 1322–1340.

14. M.T. Bordulyak, M.M. Sheremeta, Boundedness of the l-index of Laguerre-Polya entire functions, Ukrain. Mat. Zh., 55 (2003), ¹1, 91–99, (in Ukrainian); Engl. transl.: Ukrainian Math. J., 55 (2003), ¹1, 112–125.

15. M.M. Sheremeta, Generalization of the Fricke theorem on entire functions of finite index, Ukrain. Mat. Zh., 48 (1996), ¹3, 412–417; Engl. transl.: Ukrainian Math. J., 48 (1996), ¹3, 460–466.

16. B.Ya. Levin, Lectures on entire functions, American Mathematical Soc., 1996, 248 p.

17. S.N. Strochyk, M.M. Sheremeta, Analytic in the unit disc functions of bounded index, Dopov. Akad. Nauk Ukr., 1 (1993), 19–22. (in Ukrainian)

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

19. V.O. Kushnir, M.M. Sheremeta, Analytic functions of bounded l-index, Mat. Stud., 12 (1999), ¹1, 59–66.

20. M.T. Bordulyak, M.M. Sheremeta, Boundedness of the L-index of an entire function of several variables, Dopov. Akad. Nauk Ukr., 9 (1993), 10–13. (in Ukrainian)

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

22. 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.

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

Pages
27-32
Volume
47
Issue
1
Year
2017
Journal
Matematychni Studii
Full text of paper
Table of content of issue