| Title:
|
Non-homogeneous directional equations: Slice solutions belonging to functions of bounded $L$-index in the unit ball (English) |
| Author:
|
Bandura, Andriy |
| Author:
|
Salo, Tetyana |
| Author:
|
Skaskiv, Oleh |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
247-260 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a given direction ${\bf b}\in \mathbb {C}^n\setminus \{{\bf 0}\}$ we study non-homogeneous directional linear higher-order equations whose all coefficients belong to a class of joint continuous functions which are holomorphic on intersection of all directional slices with a unit ball. Conditions are established providing boundedness of $L$-index in the direction with a positive continuous function $L$ satisfying some behavior conditions in the unit ball. The provided conditions concern every solution belonging to the same class of functions as the coefficients of the equation. Our considerations use some estimates involving a directional logarithmic derivative and distribution of zeros on all directional slices in the unit ball. (English) |
| Keyword:
|
bounded index |
| Keyword:
|
bounded $L$-index in direction |
| Keyword:
|
slice function |
| Keyword:
|
holomorphic function |
| Keyword:
|
directional differential equation |
| Keyword:
|
bounded $l$-index |
| Keyword:
|
directional derivative |
| Keyword:
|
unit ball |
| MSC:
|
32A10 |
| MSC:
|
32A17 |
| MSC:
|
32A37 |
| DOI:
|
10.21136/MB.2023.0121-22 |
| . |
| Date available:
|
2024-07-10T15:06:01Z |
| Last updated:
|
2024-07-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152471 |
| . |
| Reference:
|
[1] Baksa, V., Bandura, A., Skaskiv, O.: Analogs of Hayman's theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded $L$-index in joint variables.Math. Slovaca 70 (2020), 1141-1152. Zbl 1478.32004, MR 4156814, 10.1515/ms-2017-0420 |
| Reference:
|
[2] Bandura, A., Martsinkiv, M., Skaskiv, O.: Slice holomorphic functions in the unit ball having a bounded $L$-index in direction.Axioms 10 (2021), Article ID 4, 15 pages. MR 4409465, 10.3390/axioms10010004 |
| Reference:
|
[3] Bandura, A. I., Salo, T. M., Skaskiv, O. B.: Slice holomorphic functions in the unit ball: Boundedness of $L$-index in a direction and related properties.Mat. Stud. 57 (2022), 68-78. Zbl 1487.32008, MR 4409465, 10.30970/ms.57.1.68-78 |
| Reference:
|
[4] Bandura, A., Shegda, L., Skaskiv, O., Smolovyk, L.: Some criteria of boundedness of $L$-index in a direction for slice holomorphic functions in the unit ball.Int. J. Appl. Math. 34 (2021), 775-793. MR 4409465, 10.12732/ijam.v34i4.13 |
| Reference:
|
[5] Bandura, A., Skaskiv, O.: Boundedness of the $L$-index in a direction of entire solutions of second order partial differential equation.Acta Comment. Univ. Tartu. Math. 22 (2018), 223-234. Zbl 1422.32004, MR 3911033, 10.12697/ACUTM.2018.22.18 |
| Reference:
|
[6] Bandura, A., Skaskiv, O.: Analog of Hayman's theorem and its application to some system of linear partial differential equations.J. Math. Phys. Anal. Geom. 15 (2019), 170-191. Zbl 1426.32001, MR 3968733, 10.15407/mag15.02.170 |
| Reference:
|
[7] Bandura, A., Skaskiv, O.: Linear directional differential equations in the unit ball: Solutions of bounded $L$-index.Math. Slovaca 69 (2019), 1089-1098. Zbl 1478.32003, MR 4017393, 10.1515/ms-2017-0292 |
| Reference:
|
[8] Bandura, A., Skaskiv, O.: Slice holomorphic functions in several variables with bounded $L$-index in direction.Axioms 8 (2019), Article ID 88, 12 pages. Zbl 1432.32002, 10.3390/axioms8030088 |
| Reference:
|
[9] Bandura, A., Skaskiv, O.: Some criteria of boundedness of the $L$-index in direction for slice holomorphic functions of several complex variables.J. Math. Sci., New York 244 (2020), 1-21. Zbl 1435.30138, MR 4445786, 10.1007/s10958-019-04600-7 |
| Reference:
|
[10] Bandura, A., Skaskiv, O., Filevych, P.: Properties of entire solutions of some linear PDE's.J. Appl. Math. Comput. Mech. 16 (2017), 17-28. MR 3671887, 10.17512/jamcm.2017.2.02 |
| Reference:
|
[11] Bandura, A., Skaskiv, O., Smolovyk, L.: Slice holomorphic solutions of some directional differential equations with bounded $L$-index in the same direction.Demonstr. Math. 52 (2019), 482-489. Zbl 1436.32004, MR 4044563, 10.1515/dema-2019-0043 |
| Reference:
|
[12] Bordulyak, M. T.: A proof of Sheremeta's conjecture concerning entire function of bounded $l$-index.Mat. Stud. 12 (1999), 108-110. Zbl 0974.30554, MR 1737836 |
| Reference:
|
[13] Bordulyak, M. T., Sheremeta, M. M.: On the existence of entire functions of bounded $l$-index and $l$-regular growth.Ukr. Math. J. 48 (1996), 1322-1340. Zbl 0932.30025, MR 1429603, 10.1007/BF02595355 |
| Reference:
|
[14] Fricke, G. H.: Functions of bounded index and their logarithmic derivatives.Math. Ann. 206 (1973), 215-223. Zbl 0251.30026, MR 0325962, 10.1007/BF01429209 |
| Reference:
|
[15] Fricke, G. H.: Entire functions of locally slow growth.J. Anal. Math. 28 (1975), 101-122. Zbl 0316.30020, 10.1007/BF02786809 |
| Reference:
|
[16] Hayman, W. K.: Differential inequalities and local valency.Pac. J. Math. 44 (1973), 117-137. Zbl 0248.30026, MR 0316693, 10.2140/pjm.1973.44.117 |
| Reference:
|
[17] Hural, I. M.: About some problem for entire functions of unbounded index in any direction.Mat. Stud. 51 (2019), 107-110. Zbl 1425.32004, MR 3968758, 10.15330/ms.51.1.107-110 |
| Reference:
|
[18] Kuzyk, A. D., Sheremeta, M. N.: Entire functions of bounded $l$-distribution of values.Math. Notes 39 (1986), 3-8. Zbl 0603.30034, MR 0830838, 10.1007/BF01647624 |
| Reference:
|
[19] Kuzyk, A. D., Sheremeta, M. N.: Entire functions satisfying linear differential equations.Differ. Equations 26 (1990), 1268-1273. Zbl 0732.34006, MR 1089741 |
| Reference:
|
[20] Lepson, B.: Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index.Entire Functions and Related Parts of Analysis Proceedings of Symposia in Pure Mathematics 11. AMS, Providence (1968), 298-307 \99999MR99999 0237788 . Zbl 0199.12902, MR 0237788 |
| Reference:
|
[21] MacDonnell, J. J.: Some Convergence Theorems for Dirichlet-type Series Whose Coefficients Are Entire Functions of Bounded Index: Doctoral Dissertation.Catholic University of America, Washington (1957). MR 2938858 |
| Reference:
|
[22] Nuray, F., Patterson, R. F.: Multivalence of bivariate functions of bounded index.Matematiche 70 (2015), 225-233. Zbl 1342.32006, MR 3437188, 10.4418/2015.70.2.14 |
| Reference:
|
[23] Nuray, F., Patterson, R. F.: Vector-valued bivariate entire functions of bounded index satisfying a system of differential equations.Mat. Stud. 49 (2018), 67-74. Zbl 1414.30031, MR 3841790, 10.15330/ms.49.1.67-74 |
| Reference:
|
[24] Shah, S. M.: Entire functions of bounded index.Proc. Am. Math. Soc. 19 (1968), 1017-1022. Zbl 0164.08601, MR 0237789, 10.1090/S0002-9939-1968-0237789-2 |
| Reference:
|
[25] Shah, S. M.: Entire functions satisfying a linear differential equation.J. Math. Mech. 18 (1969), 131-136. Zbl 0165.08502, MR 0227410, 10.1512/iumj.1969.18.18013 |
| Reference:
|
[26] Shah, S. M.: Entire function of bounded index.Complex Analysis Lecture Notes in Mathematics 599. Springer, Berlin (1977), 117-145. Zbl 0361.30007, MR 0457719, 10.1007/BFb0096833 |
| Reference:
|
[27] Sheremeta, M. M.: Generalization of the fricke theorem on entire functions of finite index.Ukr. Math. J. 48 (1996), 460-466. Zbl 0932.30024, MR 1408662, 10.1007/BF02378535 |
| Reference:
|
[28] Sheremeta, M.: Analytic Functions of Bounded Index.Mathematical Studies Monograph Series 6. VNTL Publishers, Lviv (1999). Zbl 0980.30020, MR 1751042 |
| Reference:
|
[29] Sheremeta, M. M.: On the $l$-index boundedness of some composition of functions.Mat. Stud. 47 (2017), 207-210. Zbl 1414.30036, MR 3733089, 10.15330/ms.47.2.207-210 |
| Reference:
|
[30] Sheremeta, M. M., Bordulyak, M. T.: Boundedness of the $l$-index of Laguerre-Pólya entire functions.Ukr. Math. J. 55 (2003), 112-125 \99999DOI99999 10.1023/A:1025076720052 . Zbl 1038.30015, MR 2034907 |
| Reference:
|
[31] Strelitz, S.: Asymptotic properties of entire transcendental solutions of algebraic differential equations.Value Distribution Theory and Its Applications Contemporary Mathematics 25. AMS, Providence (1983), 171-214. Zbl 0546.34004, MR 730048, 10.1090/conm/025 |
| . |
