The properties of solutions of the following differential equation
$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $
are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| < 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper.
Citation: Jianren Long, Pengcheng Wu, Sangui Zeng. On properties of solutions of complex differential equations in the unit disc[J]. AIMS Mathematics, 2021, 6(8): 8256-8275. doi: 10.3934/math.2021478
The properties of solutions of the following differential equation
$ f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = F(z) $
are studied, where $ A_{j}(z) $ and $ F(z) $ are analytic in the unit disc $ \mathbb{D} = \{z:|z| < 1\} $, $ j = 0, 1, \ldots, k-1 $. First, the growth of solutions of the equation is estimated. Second, some coefficient's conditions such that the solution of the equation belong to Hardy type spaces are showed. Finally, some related question are studied in this paper.
| [1] |
T. B. Cao, H. X Yi, The growth of solutions of linear differential equations with coefficients of iterated order in the unit disc, J. Math. Anal. Appl., 319 (2006), 278–294. doi: 10.1016/j.jmaa.2005.09.050
|
| [2] |
I. Chyzhykov, G. G. Gundersen, J. Heittokangas, Linear differential equations and logarithmic derivate of estimates, Proc. Lond. Math. Soc., 86 (2003), 735–754. doi: 10.1112/S0024611502013965
|
| [3] |
I. Chyzhykov, J. Heittokangas, J. Rättyä, Sharp logarithmic derivative estimates with applications to ordinary differential equations in the unit disc, J. Aust. Math. Soc., 88 (2010), 145–167. doi: 10.1017/S1446788710000029
|
| [4] | P. L. Duren, Theory of $H^{p}$ Spaces, New York-London: Academic Press, 1970. |
| [5] |
J. Gröhn, J.-M. Huusko, J. Rättyä, Linear differential equations with slowly growing solutions, Trans. Amer. Math. Soc., 370 (2018), 7201–7227. doi: 10.1090/tran/7265
|
| [6] | W. K. Hayman, Meromorphic Functions, Oxford: Clarendon Press, 1964. |
| [7] | J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1–54. |
| [8] |
J. Heittokangas, R. Korhonen, J. Rättyä, Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space, Nagoya Math. J., 187 (2007), 91–113. doi: 10.1017/S0027763000025861
|
| [9] |
J. Heittokangas, R. Korhonen, J. Rättyä, Linear differential equations with coefficients in weighted Bergman and Hardy space, Trans. Amer. Math. Soc., 360 (2008), 1035–1055. doi: 10.1090/S0002-9947-07-04335-8
|
| [10] | J. Heittokangas, R. Korhonen, J. Rättyä, Growth estimates for solutions of nonhomogeneous linear complex differential equations, Ann. Acad. Sci. Fenn. Math., 34 (2009), 145–156. |
| [11] | I. Laine, Nevanlinna Theory and Complex Differential Equations, Berlin-New York: Walter de Gruyter, 1993. |
| [12] | J. R. Long, Y. Sun, S. M. Zhang, G. M. Hu, Second-order linear differential equations with solutions in analytic function spaces, J. Funct. Space., 2019 (2019), 1–9. |
| [13] | M. X. Li, L. P. Xiao, Solutions of linear differential equations in the unit disc, J. Math. Reseach Appl., 34 (2014), 729–735. |
| [14] |
H. Li, H. Wulan, Linear differential equations with solutions in the $Q_{k}$ spaces, J. Math. Anal. Appl., 375 (2011), 478–489. doi: 10.1016/j.jmaa.2010.09.028
|
| [15] | Ch. Pommerenke, On the mean growth of the solutions of complex linear differential equations in the disk, Complex Var., 1 (1982), 23–38. |
| [16] |
Y. Sun, J. R. Long, G. M. Hu, Nonlinear complex differential equations with solutions in some function spaces, Complex Var. Ellip. Equa., 64 (2019), 300–314. doi: 10.1080/17476933.2018.1429420
|
| [17] | K. Zhu, Bloch type spaces of analytic functions, Rochy Mountain J. Math., 23 (1993), 1143–1177. |
Jianren Long, Pengcheng Wu, Sangui Zeng. On properties of solutions of complex differential equations in the unit disc[J]. AIMS Mathematics, 2021, 6(8): 8256-8275. doi: 10.3934/math.2021478
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