About some problem for entire functions of unbounded index in any direction |
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| Author |
math@nung.edu.ua
Department of Advanced Mathematics,
Ivano-Frankivsk National Technical University of Oil and Gas,
Ivano-Frankivsk, Ukraine
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| Abstract |
In this paper, we select a class of entire functions $F(z_1,\ldots,z_n)$ such that
for any direction $(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{0\}$ and
for every point $(z_1^0,\ldots, z_n^0)\in\mathbb{C}$ the function $F(z_1^0 + tb_1,\ldots, z_n^0 + tb_n)$ is of bounded
index as a function in variable $t\in\mathbb{C},$ but the function $F$ is of unbounded index in every direction
$(b_1,\ldots,b_n).$
This result partially solves Problem 18 from the article A. I. Bandura, O. B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), no.1, 103–109.
This problem concerns with existence of entire functions of unbounded $L$-index in any direction, where $L:\mathbb{C}^n\to\mathbb{R}_+$ is a continuous function
and $n\ge 3.$
Our result solves the problem in the case $L\equiv 1.$
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| Keywords |
bounded index; unbounded index in any direction; entire function; several variables
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| DOI |
doi:10.15330/ms.51.1.107-110
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| Reference |
1. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat.
Stud., 43 (2015), ¹1, 103–109. doi: 10.15330/ms.43.1.103-109
2. A.I. Bandura, A class of entire functions of unbounded index in each direction, Mat. Stud., 44 (2015), ¹1, 107–112. doi: 10.15330/ms.44.1.107-112 3. A.I. Bandura, O.B. Skaskiv, Entire bivariate functions of unbounded index in each direction, Nonlinear Oscillations, 21 (2018), ¹4, 435–443. 4. 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), ¹1-2, 8–12. (in Ukrainian) 5. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), ¹1, 30–52. (in Ukrainian) 6. 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), ¹2, 223–234. doi: 10.12697/ACUTM.2018.22.18 7. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I. E. Chyzhykov, 2016, 128 p. 8. G.H. Fricke, Functions of bounded index and their logarithmic derivatives, Math. Ann., 206 (1973), 215–223. 9. O.B. Skaskiv, Progress in the open problems in theory of functions of bounded index, Mat. Stud., 49 (2018), ¹1, 109–112. doi: 10.15330/ms.49.1.109-112 10. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), ¹1, 3–8. doi:10.1007/BF01647624 11. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307. 12. M.N. Sheremeta, A.D. Kuzyk, Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J., 33 (1992), ¹2, 304–312. doi:10.1007/BF00971102 13. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 14. F. Nuray, R.F. Patterson, Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), ¹1, 67–74, doi: 10.15330/ms.49.1.67-74 |
| Pages |
107-110
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| Volume |
51
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| Issue |
1
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| Year |
2019
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| Journal |
Matematychni Studii
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| Full text of paper | |
| Table of content of issue |