Entire curves having bounded $l$-index in $\ell_{\infty}$

A. I. Bandura

doi:10.30970/ms.52.1.108-112

In this paper we propose an approach to introduce a concept of bounded index in an infinite-dimensional space. Our object of investigation is the space $\ell^\infty$ equipped with the norm $\|x\|_{\infty }=\sup\{|x_{n}|\colon n\in\mathbb{N}\}.$ We consider entire curves from $\mathbb{C}$ to ${\ell}_{\infty}$ and prove proposition indicating connection between of the $l$-index boundedness of every component of the curve and the $l$-index boundedness of the curve. Moreover, we obtain sufficient conditions of the $l$-index boundednes of entire curves in the space. They describe local behavior of norm of derivatives of the entire curves on the discs. Also, there is posed a problem on necessary conditions of the $l$-index boundedness of entire curves in infinite-dimensional spaces.

1. Baksa V.P., Analytic vector-functions in the unit ball having bounded L-index in joint variables, Carpathian Mathematical Publications (in print).

2. Baksa V.P., Bandura A.I., Skaskiv O.B., Analogs of Frickes theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, submitted to Proceedings of IAMM of NASU.

3. Bandura A., Skaskiv O., Sufficient conditions of boundedness of L-index and analog of Haymans Theorem for analytic functions in a ball, Stud. Univ. BabeЃCs-Bolyai Math., 63 (2018), №4, 483-501. doi:10.24193/subbmath.2018.4.06

4. Bandura A.I., Skaskiv O.B. Analytic functions in the unit ball of bounded L-index: asymptotic and local properties, Mat. Stud. 48 (2017), №1, 37.73. doi: 10.15330/ms.48.1.37-73

5. Bandura A.I., Some improvements of criteria of L-index boundedness in direction, Mat. Stud., 47 (2017), №1, 27.32. doi: 10.15330/ms.47.1.27-32

6. Bandura A., Skaskiv O., Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48 (2018), №1, 103.116. doi: 10.30755/NSJOM.06997

7. Bandura A., Petrechko N., Skaskiv O., Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Haymans theorem, Mat. Bohemica, 143 (2018), №4, 339.354. doi: 10.21136/MB.2017.0110-16

8. Bandura A.I., Skaskiv O.B., Tsvigun V.L., Some characteristic properties of analytic functions in DЃ~C of bounded L-index in joint variables, Bukovyn. Mat. Zh., 6 (2018), №1.2, 21-31.

9. Bordulyak M.T., Sheremeta M.M., Boundedness of l-index of analytic curves, Mat. Stud., 36 (2011), №2, 152-161.

10. Fricke G.H., Entire functions of locally slow growth, J. Anal. Math., 28 (1975), №1, 101-122.

11. Fricke G.H., Functions of bounded index and their logarithmic derivatives, Math. Ann., 206 (1973), 215-223.

12. Heath L.F., Vector-valued entire functions of bounded index satisfying a differential equation, Journal of Research of NBS, 83B (1978), №1, 75-79.

13. Kuzyk A.D., Sheremeta M.N., Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3-8. doi:10.1007/BF01647624

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

15. Nuray F., Patterson R.F., Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), №2, 225-233. doi: 10.4418/2015.70.2.14

16. Nuray F., Patterson R.F., Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), №1, 67-74. doi: 10.15330/ms.49.1.67-74

17. Roy R., Shah S.M., Vector-valued entire functions satisfying a differential equation, J. Math. Anal. Appl., 116 (1986), №2, 349-362.

18. Roy R., Shah S.M., Growth properties of vector entire functions satisfying differential equations, Indian J. Math., 28 (1986), №1, 25-35.

19. Shah S.M., Entire function of bounded index. In: J.D. Buckholtz, T.J. Suffridge (eds.) Complex Analysis, Lecture Notes in Mathematics, V.599, 117-145, Springer, Berlin, Heidelberg, 1977.

20. Sheremeta M., Analytic functions of bounded index, Lviv: VNTL Publishers, 1999.

21. Sheremeta M., Boundedness of l .M-index of analytic curves, Visn. Lviv. Un-ty. Ser. Mech.-Math., 75 (2011), 226-231. (in Ukrainian)

22. Sheremeta M.N., Kuzyk A.D., Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J., 33 (1992), №2, 304-312. doi:10.1007/BF00971102

	Entire curves having bounded $l$-index in $\ell_{\infty}$
Author	A. I. Bandura andriykopanytsia@gmail.com Department of Advanced Mathematics Ivano-Frankivsk National Technical University of Oil and Gas Ivano-Frankivsk, Ukraine
Abstract	In this paper we propose an approach to introduce a concept of bounded index in an infinite-dimensional space. Our object of investigation is the space $\ell^\infty$ equipped with the norm $\\|x\\|_{\infty }=\sup\{\|x_{n}\|\colon n\in\mathbb{N}\}.$ We consider entire curves from $\mathbb{C}$ to ${\ell}_{\infty}$ and prove proposition indicating connection between of the $l$-index boundedness of every component of the curve and the $l$-index boundedness of the curve. Moreover, we obtain sufficient conditions of the $l$-index boundednes of entire curves in the space. They describe local behavior of norm of derivatives of the entire curves on the discs. Also, there is posed a problem on necessary conditions of the $l$-index boundedness of entire curves in infinite-dimensional spaces.
Keywords	bounded index; bounded $l$-index; entire curve; $\ell_{\infty}$
DOI	doi:10.30970/ms.52.1.108-112
Reference	1. Baksa V.P., Analytic vector-functions in the unit ball having bounded L-index in joint variables, Carpathian Mathematical Publications (in print). 2. Baksa V.P., Bandura A.I., Skaskiv O.B., Analogs of Frickes theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables, submitted to Proceedings of IAMM of NASU. 3. Bandura A., Skaskiv O., Sufficient conditions of boundedness of L-index and analog of Haymans Theorem for analytic functions in a ball, Stud. Univ. BabeЃCs-Bolyai Math., 63 (2018), №4, 483-501. doi:10.24193/subbmath.2018.4.06 4. Bandura A.I., Skaskiv O.B. Analytic functions in the unit ball of bounded L-index: asymptotic and local properties, Mat. Stud. 48 (2017), №1, 37.73. doi: 10.15330/ms.48.1.37-73 5. Bandura A.I., Some improvements of criteria of L-index boundedness in direction, Mat. Stud., 47 (2017), №1, 27.32. doi: 10.15330/ms.47.1.27-32 6. Bandura A., Skaskiv O., Asymptotic estimates of entire functions of bounded L-index in joint variables, Novi Sad J. Math., 48 (2018), №1, 103.116. doi: 10.30755/NSJOM.06997 7. Bandura A., Petrechko N., Skaskiv O., Maximum modulus in a bidisc of analytic functions of bounded L-index and an analogue of Haymans theorem, Mat. Bohemica, 143 (2018), №4, 339.354. doi: 10.21136/MB.2017.0110-16 8. Bandura A.I., Skaskiv O.B., Tsvigun V.L., Some characteristic properties of analytic functions in DЃ~C of bounded L-index in joint variables, Bukovyn. Mat. Zh., 6 (2018), №1.2, 21-31. 9. Bordulyak M.T., Sheremeta M.M., Boundedness of l-index of analytic curves, Mat. Stud., 36 (2011), №2, 152-161. 10. Fricke G.H., Entire functions of locally slow growth, J. Anal. Math., 28 (1975), №1, 101-122. 11. Fricke G.H., Functions of bounded index and their logarithmic derivatives, Math. Ann., 206 (1973), 215-223. 12. Heath L.F., Vector-valued entire functions of bounded index satisfying a differential equation, Journal of Research of NBS, 83B (1978), №1, 75-79. 13. Kuzyk A.D., Sheremeta M.N., Entire functions of bounded l-distribution of values, Math. Notes, 39 (1986), №1, 3-8. doi:10.1007/BF01647624 14. Lepson B., Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math., 2 (1968), 298-307. 15. Nuray F., Patterson R.F., Multivalence of bivariate functions of bounded index, Le Matematiche, 70 (2015), №2, 225-233. doi: 10.4418/2015.70.2.14 16. Nuray F., Patterson R.F., Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations, Mat. Stud., 49 (2018), №1, 67-74. doi: 10.15330/ms.49.1.67-74 17. Roy R., Shah S.M., Vector-valued entire functions satisfying a differential equation, J. Math. Anal. Appl., 116 (1986), №2, 349-362. 18. Roy R., Shah S.M., Growth properties of vector entire functions satisfying differential equations, Indian J. Math., 28 (1986), №1, 25-35. 19. Shah S.M., Entire function of bounded index. In: J.D. Buckholtz, T.J. Suffridge (eds.) Complex Analysis, Lecture Notes in Mathematics, V.599, 117-145, Springer, Berlin, Heidelberg, 1977. 20. Sheremeta M., Analytic functions of bounded index, Lviv: VNTL Publishers, 1999. 21. Sheremeta M., Boundedness of l .M-index of analytic curves, Visn. Lviv. Un-ty. Ser. Mech.-Math., 75 (2011), 226-231. (in Ukrainian) 22. Sheremeta M.N., Kuzyk A.D., Logarithmic derivative and zeros of an entire function of bounded l-index, Sib. Math. J., 33 (1992), №2, 304-312. doi:10.1007/BF00971102
Pages	108-112
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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