Sufficient conditions of boundedness of L-index in joint variables |
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| Author |
andriykopanytsia@gmail.com; mbordulyak@yahoo.com, olskask@gmail.com
Department of Higher Mathematics
Ivano-Frankivsk National Technical University of Oil and Gas; Department of Function Theory and Theory of Probability
Ivan Franko National University of Lviv
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| Abstract |
A concept of boundedness of $\mathbf{L}$-index in joint variables (see in Bordulyak M.T. The space of entire in $\mathbb{C}^n$ functions of bounded $L$-index, Mat. Stud., 4 (1995), 53-58. (in Ukrainian)) is generalised for $\mathbf{L}(z)=(l_1(z),$ $\ldots,$ $l_{n}(z)),$
$z\in\mathbb{C}^n.$ We proved criteria of boundedness of $\mathbf{L}$-index in joint variables
and established a connection between the classes of entire functions of bounded $l_j$-index in each direction $\mathbf{e}_j$ and functions of bounded $\mathbf{L}$-index in joint variables.
We deduce new sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables. The obtained restrictions describe the behaviour of logarithmic derivative in each variable and the distribution of zeros.
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| Keywords |
entire function; bounded L-index in joint variables; bounded L-index in direction; directional
logarithmic derivative; distibution of zeros
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| DOI |
doi:10.15330/ms.45.1.12-26
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| Reference |
1. 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)
2. M.T. Bordulyak, The space of entire in Cn functions of bounded L-index// Mat. Stud., 4 (1995), 53–58. (in Ukrainian) 3. Bordulyak, M. T. Boundedness of L-index of entire functions of several complex variables: diss. ... Cand. Phys. and Math. Sciences, Lviv University, Lviv, 1996, 100 p. (in Ukrainian) 4. A.D. Kuzyk, M.M. Sheremeta, Entire functions of bounded l-distribution of values, Math. notes, 39 (1986), ¹1, 3–8. 5. M. Sheremeta, Analytic functions of bounded index, Lviv: VNTL Publishers, 1999, 141 p. 6. F. Nuray, R.F. Patterson, Entire bivariate functions of exponential type, Bull. Math. Sci., 5 (2015), ¹2, 171–177. 7. F. Nuray, R.F. Patterson, Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), ¹2, 225–233. 8. M. Salmassi, Functions of bounded indices in several variables, Indian J. Math., 31 (1989), ¹3, 249–257. 9. Gopala J. Krishna, S.M. Shah Functions of bounded indices in one and several complex variables, In: Mathematical essays dedicated to A.J. Macintyre, Ohio Univ. Press, Athens, Ohio, 1970, 223–235. 10. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded L-index in direction, Mat. Stud., 27 (2007), ¹1, 30–52. (in Ukrainian) 11. A.I. Bandura, O.B. Skaskiv, Entire functions of bounded and unbounded index in direction, Mat. Stud., 27 (2007), ¹2, 211–215. (in Ukrainian) 12. A.I. Bandura, O.B. Skaskiv, Sufficient sets for boundedness L-index in direction for entire functions, Mat. Stud., 30 (2008), ¹2, 177–182. 13. A.I. Bandura, On boundedness of the L-index in the direction for entire functions with plane zeros, Mat. Visn. Nauk. Tov. Im. Shevchenka, 6 (2009), 44–49. (in Ukrainian) 14. A.I. Bandura, O.B. Skaskiv, Open problems for entire functions of bounded index in direction, Mat. Stud., 43 (2015), ¹1, 103–109. 15. B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., Amer. Math. Soc.: Providence, Rhode Island, 2 (1968), 298–307. |
| Pages |
12-26
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| Volume |
45
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| Issue |
1
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| Year |
2016
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| Journal |
Matematychni Studii
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| Full text of paper | |
| Table of content of issue |