# A class of entire functions of unbounded index in each direction

Author
Department of higher mathematics, Ivano-Frankivsk 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, 103--109.
Keywords
entire functions of two variables; unbounded index in direction; directional derivative
DOI
doi:10.15330/ms.44.1.107-112
Reference
1. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index 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 L-index in direction and of bounded L-index 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 l-index, Izv. Vyssh. Uchebn. Zaved. Mat., 9 (1992), 81–87 (in Russian); translation in Russian Math. (Iz. VUZ) 36 (1992), ¹9, 76–82.

Pages
107-112
Volume
44
Issue
1
Year
2015
Journal
Matematychni Studii
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