Growth of \(\phi \)-order solutions of linear differential equations with meromorphic coefficients on the complex plane. (English) Zbl 1460.34108

Kara, Mohamed Abdelhak; Belaïdi, Benharrat

Ural Math. J. 6, No. 1, 95-113 (2020).

In this paper, the authors study the growth of higher order linear differential equations with meromorphic coefficients of \(\varphi\)-order on the comlpex plane. They prove many theorems by using the concepts of \(\varphi\)-order and \(\varphi\)-type. These theorems extend previous results due to Chyzhykov, Semochko, Belaïdi, Cao, Xu, Chen and Kinnunen. This work is interesting.

Reviewer: Karima Hamani (Mostaganem)

MSC:

34M10	Oscillation, growth of solutions to ordinary differential equations in the complex domain
34M03	Linear ordinary differential equations and systems in the complex domain

Keywords:

linear differential equations; entire function; meromorphic function; \(\phi\)-order; \(\phi\)-type

References:

[1]	Belaïdi B., “Fast growing solutions to linear differential equations with entire coefficients having the same \(\rho_{\varphi }\)-order”, J. Math. Appl., 42 (2019), 63-77 · Zbl 1425.34104
[2]	Belaïdi B., “Growth of \(\rho_{\varphi }\)-order solutions of linear differential equations with entire coefficients”, Pan-American Math. J., 27:4 (2017), 26-42
[3]	Bela\"{i}di B., “Growth and oscillation of solutions to linear differential equations with entire coefficients having the same order”, Electron. J. Differential Equations., 2009, no. 70, 1- 10 · Zbl 1170.34356
[4]	Bernal L. G., “On growth k-order of solutions of a complex homogeneous linear differential equation”, Proc. Amer. Math. Soc., 101:2 (1987), 317- 322 · Zbl 0652.34008
[5]	Cao T.-B., Xu J. F., Chen Z. X., “On the meromorphic solutions of linear differential equations on the complex plane”, J. Math. Anal. Appl., 364:1 (2010), 130- 142 · Zbl 1194.34161
[6]	Chiang Y.-M., Hayman W. K., “Estimates on the growth of meromorphic solutions of linear differential equations”, Comment. Math. Helv., 79:3 (2004), 451- 470 · Zbl 1057.34110
[7]	Chyzhykov I., Semochko N., “Fast growing entire solutions of linear differential equations”, Math. Bull. Shevchenko Sci. Soc., 13 (2016), 68- 83 · Zbl 1374.34362
[8]	Frank G., Hellerstein S., “On the meromorphic solutions of non-homogeneous linear differential equations with polynomial coefficients”, Proc. London Math. Soc., 53:3 (1986), 407- 428 · Zbl 0635.34005
[9]	Gundersen G. G., “Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates”, J. London Math. Soc., 37:1 (1988), 88- 104 · Zbl 0638.30030
[10]	Hayman W. K., Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964, 191 pp. · Zbl 0115.06203
[11]	Juneja O. P., Kapoor G. P., Bajpai S. K., “On the \((p, q)\)-order and lower \((p, q)\)-order of an entire function”, J. Reine Angew. Math., 282 (1976), 53- 67 · Zbl 0321.30031
[12]	Kara M. A., Bela\"{i}di B., “Some estimates of the \(\phi \)-order and the \(\phi \)-type of entire and meromorphic functions”, Int. J. Open Problems Complex Analysis, 10:3 (2019), 42- 58
[13]	Kinnunen L., “Linear differential equations with solutions of finite iterated order”, Southeast Asian Bull. Math., 22:4 (1998), 385- 405 · Zbl 0934.34076
[14]	Laine I., Nevanlinna Theory and Complex Differential Equations, De Gruyter Studies in Mathematics, 15, Walter de Gruyter & Co., Berlin, 1993, 341 pp. · Zbl 0784.30002
[15]	Li L. M., Cao T. B., “Solutions for linear differential equations with meromorphic coefficients of \([p, q]\)-order in the plane”, Electron. J. Differential Equations, 2012, no. 195, 1- 15 · Zbl 1320.53047
[16]	Liu J., Tu J., Shi L.Ż., “Linear differential equations with entire coefficients of \([p, q]\)-order in the complex plane”, J. Math. Anal. Appl., 372 (2010), 55- 67 · Zbl 1209.34110
[17]	Mulyava O. M., Sheremeta M. M., Trukhan Yu. S., “Properties of solutions of a heterogeneous differential equation of the second order”, Carpathian Math. Publ., 11:2 (2019), 379- 398 · Zbl 1435.30011
[18]	Nevanlinna R., “Zur theorie der meromorphen funktionen”, Acta Math., 46:1-2 (1925), 1- 99 (in German) · JFM 51.0254.05
[19]	Seneta E., Regularly Varying Functions, Lecture Notes in Math., 508, Springer-Verlag, Berlin-Heidelberg, 1976, 116 pp. · Zbl 0324.26002
[20]	Sheremeta M. N., “Connection between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power series expansion”, Izv. Vyssh. Uchebn. Zaved. Mat., 2 (1967), 100-108 (in Russian)
[21]	Tu J., Chen Z.-X., “Growth of solutions of complex differential equations with meromorphic coefficients of finite iterated order”, Southeast Asian Bull. Math., 33:1 (2009), 153-164 · Zbl 1212.34283
[22]	Yang L., Value Distribution Theory, Springer-Verlag, Berlin-Heidelberg, 1993, 269 pp. · Zbl 0790.30018