Linear directional differential equations in the unit ball: solutions of bounded L-index
-
Andriy Bandura
and
Oleh Skaskiv
Abstract
We study sufficient conditions of boundedness of L-index in a direction b ∈ ℂn ∖ {0} for analytic solutions in the unit ball of a linear higher order non-homogeneous differential equation with directional derivatives. These conditions are restrictions by the analytic coefficients in the unit ball of the equation. Also we investigate asymptotic behavior of analytic functions of bounded L-index in the direction and estimate its growth. The results are generalizations of known propositions for entire functions of several variables.
(Communicated by Michal Fečkan)
References
[1] Bandura, A.—Skaskiv, O.: Functions analytic in a unit ball of bounded L-index in joint variables, J. Math. Sci. 227 (2017), 1–12.10.1007/s10958-017-3570-6Search in Google Scholar
[2] Bandura, A. I.—Skaskiv, O. B.: Analytic functions in the unit ball of boundedL-index: asymptotic and local properties, Mat. Stud. 48 (2017), 37–73.10.15330/ms.48.2.180-188Search in Google Scholar
[3] Bandura, A. I.—Skaskiv, O. B.: Entire functions of bounded L-index in direction, Mat. Stud. 27 (2007), 30–52, (in Ukrainian).Search in Google Scholar
[4] Bandura, A. I.—Skaskiv, O. B.: Growth of entire functions of bounded L-index in direction, J. Math. Sci. 240 (2019), 21–33.10.1007/s10958-019-04333-7Search in Google Scholar
[5] Bandura, A.—Skaskiv, O.: Entire Functions of Several Variables of Bounded Index, Publisher I. E. Chyzhykov, Chyslo, Lviv, 2016.Search in Google Scholar
[6] Bandura, A. I.—Skaskiv, O. B.: Directional logarithmic derivative and the distribution of zeros of an entire function of bounded L-index along the direction, Ukrain. Mat. J. 69 (2017), 500–508.10.1007/s11253-017-1377-8Search in Google Scholar
[7] Bandura, A.—Skaskiv, O.—Filevych, P.: Properties of entire solutions of some linear PDE's, J. Appl. Math. Comput. Mech. 16 (2017), 17–28.10.17512/jamcm.2017.2.02Search in Google Scholar
[8] Bandura, A.—Skaskiv, O.: Analytic in the unit ball functions of bounded L-index in direction, available at https://arxiv.org/abs/1501.04166.10.1216/RMJ-2019-49-4-1063Search in Google Scholar
[9] Bandura, A.—Skaskiv, O.: Functions analytic in the unit ball having bounded L-index in a direction, Rocky Mountain J. Math. 49 (2019), 1063–1092.10.1216/RMJ-2019-49-4-1063Search in Google Scholar
[10] Bandura, A.—Skaskiv, O.: Entire functions of boundedL-Index: its zeros and behavior of partial logarithmic derivatives, J. Complex Anal. (2017), Art. ID 3253095, 10 pp.10.1155/2017/3253095Search in Google Scholar
[11] Fricke, G.H.—Shah, S.M.: Entire functions satisfying a linear differentialequation, Indag. Math. 78 (1975), 39–41.10.1016/1385-7258(75)90012-8Search in Google Scholar
[12] Fricke, G. H.—Shah, S. M.: On bounded value distribution and bounded index, Nonlinear Anal. 2 (1978), 423–435.10.1016/0362-546X(78)90049-4Search in Google Scholar
[13] Hayman, W. K.: Differential inequalities and local valency, Pacific J. Math. 44 (1973), 117–137.10.2140/pjm.1973.44.117Search in Google Scholar
[14] Kushnir, V. O.—Sheremeta, M. M.: Analytic functions of bounded l-index, Mat. Stud. 12 (1999), 59–66.Search in Google Scholar
[15] Kuzyk, A. D.—Sheremeta, M. M.: Entire functions of bounded l-distribution of values, Math. Notes 39 (1986), 3–8.10.1007/BF01647624Search in Google Scholar
[16] Kuzyk, A. D.—Sheremeta, M. N.: On entire functions, satisfying linear differential equations, Diff. Equations 26 (1990), 1716–1722, (in Russian).Search in Google Scholar
[17] Lepson, B.: Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math. 2 (1968), 298–307.10.1090/pspum/011/0237788Search in Google Scholar
[18] Nuray, F.—Patterson, R. F.: Entire bivariate functions of exponential type, Bull. Math. Sci. 5 (2015), 171–177.10.1007/s13373-015-0066-xSearch in Google Scholar
[19] Nuray, F.—Patterson, R. F.: Multivalence of bivariate functions of bounded index, Le Matematiche 70 (2015), 225–233.Search in Google Scholar
[20] Patterson, R.—Nuray, F.: A characterization of holomorphic bivariate functions of bounded index, Math. Slovaca 67 (2017), 731–736.10.1515/ms-2017-0005Search in Google Scholar
[21] Rudin, W.: Function Theory in the Unit Ball of ℂn, Reprint of the 1980 Edition, Springer, 2008.10.1007/978-3-540-68276-9Search in Google Scholar
[22] Shah, S. M.: Entire functions of bounded index, Proc. Amer. Math. Soc. 19 (1968), 1017–1022.10.1007/BFb0096833Search in Google Scholar
[23] Shah, S. M.: Entire functions satisfying a linear differential equation, Indiana Univ. Math. J. 18 (1969), 131–136.10.1512/iumj.1969.18.18013Search in Google Scholar
[24] Sheremeta, M.: Analytic Functions of Bounded Index, VNTL Publishers, Lviv, 1999.Search in Google Scholar
[25] Strochyk, S. N.—Sheremeta, M. M.: Analytic in the unit disc functions of bounded index, Dopov. Nats. Akad. Nauk Ukr. Mat. 1 (1993), 19–22, (in Ukrainian).Search in Google Scholar
[26] Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Math., Springer, New York, 2005.Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
Articles in the same Issue
- Regular papers
- Professor Anatolij Dvurečenskij
- A generating theorem of punctured surface triangulations with inner degree at least 4
- Natural extension of EF-spaces and EZ-spaces to the pointfree context
- On weak isometries in directed groups
- On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
- On the extensions of discrete valuations in number fields
- On certain classes of generalized derivations
- Nilpotent elements in medial semigroups
- Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
- Sharp coefficient bounds for starlike functions associated with the Bell numbers
- Sufficient conditions for Carathéodory functions and applications to univalent functions
- Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
- Linear directional differential equations in the unit ball: solutions of bounded L-index
- On oscilatory fourth order nonlinear neutral differential equations – IV
- Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
- Lerch’s theorem on nabla time scales
-
- Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
- Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
- On the amenability of a class of Banach algebras with application to measure algebra
- A bivariate Kumaraswamy-exponential distribution with application
- Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
- Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
