Sum of entire functions of bounded L-index in direction

A. I. Bandura

doi:10.15330/ms.45.2.149-158

It is proved that an entire function $F$ has bounded $L$-index in a direction $\mathbf{b}$ in arbitrary bounded domain $G$ under the assumption that $F$ does not equal identically zero on the slice $\{z^0+t\mathbf{b}: \ t\in\mathbb{C}\}$ for all $z^0\in G.$ Also it is obtained sufficient conditions of boundedness of $L$-index in direction for the sum of entire functions. They are new for entire functions of bounded $l$-index of one complex variable too. As a corollary, a class of entire functions of strongly bounded $L$-index in a direction matches with a class of entire functions of bounded $L$-index in the same direction. Moreover, we gave a negative answer to the question of Prof. S. Yu. Favorov: whether it is possible in theory of bounded $L$-index in direction to replace the assumption that $F$ is holomorphic in $\mathbb{C}^n$ by the assumption that $F$ is holomorphic on every slice $\{z^0+t\mathbf{b}: \ t\in\mathbb{C}\}$ for all $z^0\in \mathbb{C}^n.$

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

2. A.I. Bandura, O.B. Skaskiv, Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), №2, 177–182.

3. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109.

4. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26.

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

6. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3–8.

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

8. A.I. Bandura, Product of two entire functions of bounded L-index in direction is a function with the same class, Bukovyn. Mat. Zh. (in Ukrainian, accepted)

9. W.J. Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc., 22 (1969), 319–323.

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

	Sum of entire functions of bounded L-index in direction
Author	A. I. Bandura andriykopanytsia@gmail.com Ivano-Frankivsk National Technical University of Oil and Gas
Abstract	It is proved that an entire function $F$ has bounded $L$-index in a direction $\mathbf{b}$ in arbitrary bounded domain $G$ under the assumption that $F$ does not equal identically zero on the slice $\{z^0+t\mathbf{b}: \ t\in\mathbb{C}\}$ for all $z^0\in G.$ Also it is obtained sufficient conditions of boundedness of $L$-index in direction for the sum of entire functions. They are new for entire functions of bounded $l$-index of one complex variable too. As a corollary, a class of entire functions of strongly bounded $L$-index in a direction matches with a class of entire functions of bounded $L$-index in the same direction. Moreover, we gave a negative answer to the question of Prof. S. Yu. Favorov: whether it is possible in theory of bounded $L$-index in direction to replace the assumption that $F$ is holomorphic in $\mathbb{C}^n$ by the assumption that $F$ is holomorphic on every slice $\{z^0+t\mathbf{b}: \ t\in\mathbb{C}\}$ for all $z^0\in \mathbb{C}^n.$
Keywords	entire function; bounded L-index in direction; strongly bounded L-index in direction; sum of entire function; holomorphy in slice; bounded domain
DOI	doi:10.15330/ms.45.2.149-158
Reference	1. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), №1, 30–52. (in Ukrainian) 2. A.I. Bandura, O.B. Skaskiv, Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), №2, 177–182. 3. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), №1, 103–109. 4. A.I. Bandura, M.T. Bordulyak, O.B. Skaskiv, Sufficient conditions of boundedness of L-index in joint variables, Mat. Stud., 45 (2016), №1, 12–26. 5. A. Bandura, O. Skaskiv, Entire functions of several variables of bounded index, Lviv: Publisher I.E. Chyzhykov, 2016, 128 p. 6. A.D. Kuzyk, M.N. Sheremeta, Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3–8. 7. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298–307. 8. A.I. Bandura, Product of two entire functions of bounded L-index in direction is a function with the same class, Bukovyn. Mat. Zh. (in Ukrainian, accepted) 9. W.J. Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc., 22 (1969), 319–323. 10. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p.
Pages	149-158
Volume	45
Issue	2
Year	2016
Journal	Matematychni Studii
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Table of content of issue	html