A class of entire functions of unbounded index in each direction 

Author 
andriykopanytsia@gmail.com
Department of higher mathematics,
IvanoFrankivsk National Technical University of Oil and Gas

Abstract 
We select a class of entire functions $f(z_1,z_2)$ with the property
$\forall \mathbf{b}=(b_1,b_2)\in\mathbb{C}^2\setminus\{0\}$
$\forall~z_1^0,$ $z_2^0\in\mathbb{C}$ the function
$f(z_1^0+tb_1,z_2^0+tb_2)$ is of bounded index as a function in
variable $t\in\mathbb{C},$ but $f(z_1,z_2)$ is of unbounded index in
every direction $\mathbf{b}.$ Thus, it solves Problem 17 from the
article A. I. Bandura, O. B. Skaskiv, {\it Open problems for entire
functions of bounded index in direction,}
Mat. Stud., \textbf{43} (2015), no.1, 103109.

Keywords 
entire functions of two variables; unbounded index in direction; directional derivative

DOI 
doi:10.15330/ms.44.1.107112

Reference 
1. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded Lindex in direction, Mat. Stud., 27 (2007), ¹1,
30–52. (in Ukrainian)
2. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), ¹2, 211–215. (in Ukrainian) 3. A.I. Bandura, Entire function of unbounded index in any real direction, Precarpathian bulletin SSS, 1(29) (2015), 24–30. 4. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), ¹1, 103–109. dx.doi.org/10.15330/ms.43.1.103–109. 5. A.I. Bandura, O.B. Skaskiv, Entire functions of several variables of bounded Lindex in direction and of bounded Lindex in joint variables. https://arxiv.org/abs/1508.07486. 6. W.K. Hayman, Differential equations and local valency, Pac. J. Math., 44 (1973), ¹1, 117–137. 7. A.D. Kuzyk, M.N. Sheremeta, On entire functions, satisfying linear differential equations, Diff. equations, 26 (1990), ¹10, 1716–1722. (in Russian) 8. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., V.2., Amer. Math. Soc.: Providence, Rhode Island, 1968, 298–307. 9. M.N. Sheremeta, Entire functions and Dirichlet series of bounded lindex, Izv. Vyssh. Uchebn. Zaved. Mat., 9 (1992), 81–87 (in Russian); translation in Russian Math. (Iz. VUZ) 36 (1992), ¹9, 76–82. 
Pages 
107112

Volume 
44

Issue 
1

Year 
2015

Journal 
Matematychni Studii

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