In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.
Citation: Yu Chen, Guantie Deng. Fast growth and fixed points of solutions of higher-order linear differential equations in the unit disc[J]. AIMS Mathematics, 2021, 6(10): 10833-10845. doi: 10.3934/math.2021629
In this paper, we investigate the fast growing solutions of higher-order linear differential equations where $ A_0 $, the coefficient of $ f $, dominates other coefficients near a point on the boundary of the unit disc. We improve the previous results of solutions of the equations where the modulus of $ A_{0} $ is dominant near a point on the boundary of the unit disc, and obtain extensive version of iterated order of solutions of the equations where the characteristic function of $ A_{0} $ is dominant near the point. We also obtain a general result of the iterated exponent of convergence of the fixed points of the solutions of higher-order linear differential equations in the unit disc. This work is an extension and an improvement of recent results of Hamouda and Cao.
[1] | W. K. Hayman, Meromorphic functions, Oxford: Clarendon Press, 1964. |
[2] | L. Yang, Value distribution theory, Berlin: Springer-Verlag, 1993. |
[3] | I. Laine, Nevanlinna theory and complex differential equations, Berlin, New York: Walter de Gruyter, 1993. |
[4] | J. Heittokangas, On complex differential equations in the unit disc, Ann. Acad. Sci. Fenn. Math. Diss., 122 (2000), 1–54. |
[5] | T. B. Cao, The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc, J. Math. Anal. Appl., 352 (2009), 739–748. doi: 10.1016/j.jmaa.2008.11.033 |
[6] | J. Heittokangas, R. Korhonen, J. Rättyä, Fast growing solutions of linear differential equations in the unit disc, Result. Math., 49 (2006), 265–278. doi: 10.1007/s00025-006-0223-3 |
[7] | M. Tsuji, Potential theory in modern function theory, New York: Chelsea, 1975. |
[8] | I. E. Chyzhykov, G. G. Gundersen, J. Heittokangas, Linear differential equations and logarithmic derivative estimates, P. Lond. Math. Soc., 86 (2003), 735–754. doi: 10.1112/S0024611502013965 |
[9] | T. B. Cao, H. Y. 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 |
[10] | S. Hamouda, Iterated order of solutions of linear differential equations in the unit disc, Comput. Meth. Funct. Th., 13 (2013), 545–555. doi: 10.1007/s40315-013-0034-y |
[11] | B. Belaïdi, Oscillation of fast growing solutions of linear differential equations in the unit disc, Acta Univ. Sapientiae Math., 2 (2010), 25–38. |
[12] | J. Heittokangas, R. Korhonen, J. Rättyä, Growth estimates for solutions of linear complex differential equations, Ann. Acad. Sci. Fenn. Math., 29 (2004), 233–246. |
[13] | T. B. Cao, Z. S. Deng, Solutions of non-homogeneous linear differential equations in the unit disc, Ann. Pol. Math., 97 (2010), 51–61. doi: 10.4064/ap97-1-4 |