Some properties of meromorphic solutions of linear differential
equation with meromorphic coefficients

A. Z. Mokhon’ko; L. I. Kolyasa

doi:10.30970/ms.52.2.166-172

Estimations of growth of the meromorphic solutions of linear differential equations with meromorphic coefficients in terms of Nevanlinna's characteristics have been obtained; namely, it is proven that if in the equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots+a_{s+1}(z)f^{s+1}+\ldots+a_{0}(z)f=0$ the coefficients $a_{j}(z),$ $j=0,1,\ldots,n-1,$ are meromorphic functions in $\mathbb{C},$ such that the coefficients $a_{j}(z),$ $j=s+1,s+2,\ldots,n-1,$ grow slower than the coefficients $a_{s}$ do, then the equation can have at most $s$ linearly independent meromorphic solutions, the growth of which is restricted by the growth of the coefficient $a_{s}.$

1. L.G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equation, Proc. Amer. Math. Soc., 101 (1987), №2, 317-322.

2. I.E. Chyzhykov, N.S. Semochko, Fast growing entire solutions of linear differential equations, Matematychnyi Visnyk NTSh, 13 (2016), 68-83.

3. M. Frei, Uber die Losungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment. math. helv., 35 (1961), 201-222.

4. A.A. Goldberg, I.V. Ostrovskiy, Value distribution of meromorphic functions,, Transl. of Math. Monogr., V.236, Amer. Math. Soc., 2008.

5. W.K. Hayman, Meromorphic functions, Oxford, Clarendon Press, 1964.

6. J. Lin, J. Tu, L.Z. Shi, Linear differential equations with entire coefficients of [p; q]-order in the complex plane, J. Math. Anal. Appl., 372 (2010), 55-67.

7. A.Z. Mokhonko, A.A. Mokhonko, On the order of growth of the solutions of linear differential equations in the vicinity of a branching point, Ukrainskiy matematicheskiy zhurnal, 67 (2015), №1, 139-144.

8. A.Z. Mokhonko, V.D. Mokhonko, Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations, Sib. Math. J., 15 1974, 921-934. (in Russian)

9. N. Steinmetz, Nevanlinna theory, normal families, and algebraic differential equations, Springer International Publishing AG, 2017.

	Some properties of meromorphic solutions of linear differential equation with meromorphic coefficients
Author	A. Z. Mokhon’ko¹, L. I. Kolyasa² kolyasa.lubov@gmail.com¹ Lviv Polytechnic National University, Lviv, Ukraine
Abstract	Estimations of growth of the meromorphic solutions of linear differential equations with meromorphic coefficients in terms of Nevanlinna's characteristics have been obtained; namely, it is proven that if in the equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots+a_{s+1}(z)f^{s+1}+\ldots+a_{0}(z)f=0$ the coefficients $a_{j}(z),$ $j=0,1,\ldots,n-1,$ are meromorphic functions in $\mathbb{C},$ such that the coefficients $a_{j}(z),$ $j=s+1,s+2,\ldots,n-1,$ grow slower than the coefficients $a_{s}$ do, then the equation can have at most $s$ linearly independent meromorphic solutions, the growth of which is restricted by the growth of the coefficient $a_{s}.$
Keywords	linear differential equation; meromorphic function; entire function; order of growth
DOI	doi:10.30970/ms.52.2.166-172
Reference	1. L.G. Bernal, On growth k-order of solutions of a complex homogeneous linear differential equation, Proc. Amer. Math. Soc., 101 (1987), №2, 317-322. 2. I.E. Chyzhykov, N.S. Semochko, Fast growing entire solutions of linear differential equations, Matematychnyi Visnyk NTSh, 13 (2016), 68-83. 3. M. Frei, Uber die Losungen linearer Differentialgleichungen mit ganzen Funktionen als Koeffizienten, Comment. math. helv., 35 (1961), 201-222. 4. A.A. Goldberg, I.V. Ostrovskiy, Value distribution of meromorphic functions,, Transl. of Math. Monogr., V.236, Amer. Math. Soc., 2008. 5. W.K. Hayman, Meromorphic functions, Oxford, Clarendon Press, 1964. 6. J. Lin, J. Tu, L.Z. Shi, Linear differential equations with entire coefficients of [p; q]-order in the complex plane, J. Math. Anal. Appl., 372 (2010), 55-67. 7. A.Z. Mokhonko, A.A. Mokhonko, On the order of growth of the solutions of linear differential equations in the vicinity of a branching point, Ukrainskiy matematicheskiy zhurnal, 67 (2015), №1, 139-144. 8. A.Z. Mokhonko, V.D. Mokhonko, Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations, Sib. Math. J., 15 1974, 921-934. (in Russian) 9. N. Steinmetz, Nevanlinna theory, normal families, and algebraic differential equations, Springer International Publishing AG, 2017.
Pages	166-172
Volume	52
Issue	2
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some properties of meromorphic solutions of linear differential equation with meromorphic coefficients